Difference Lindelöf Perfect Function in Topology and Statistical Modeling

Abstract

We develop the theory of Difference Lindelöf perfect functions. Through difference covers, we provide intrinsic characterizations; prove stability under composition, subspace restriction, and suitable products; and obtain preservation theorems. Under standard separation axioms, properties such as D-countable compactness, regularity, paracompactness, and the closedness of projections transfer along D-Lindelöf perfect maps. We also connect the framework to statistics. Uses include decision regions expressed as differences of open sets and parameter screening, with visualizations of countable subcovers and their pushforwards. The results point to practical countable cores for learning and inference and suggest extensions to bitopological and fuzzy contexts.

1. Introduction

In general topology, open sets underpin the creation of new classes of sets and the characterization of essential topological properties. Within this framework, Tong [1] defined difference sets (D-sets) from open sets and formulated separation axioms $D_i$ ($i=0,1,2$). Later refinements advanced these ideas: Caldas [2] used semi-open sets to define $s\text{−}{D}_i$ spaces, and Jafari [3] introduced $p\text{−}D$-sets using p-open sets. Additional generalizations include the $\kappa \text{−}{D}_i$ spaces defined by Caldas et al. [4] and the D-continuity decompositions of Ekici and Jafari [5]. Keskin [6] subsequently defined $bD$-sets as the difference of two b-open sets and used them to obtain weak separation axioms, while Balasubramanian [7] presented a new generalization of separation axioms to $g\text{−}{D}_i$ spaces. In subsequent work by Balasubramanian and Lakshmi [8], $gpr\text{−}{D}_i$ spaces were defined using $gpr\text{−}sets$; Sreeja and Janaki introduced new generalized $\pi gb$-separation axioms [9]; and Gnanachandra and Thangavelu proposed $pgpr\text{−}D$-sets [10], broadening the area. Recent contributions include Padma et al.’s $Q^{*}D$-sets [11] and the $b^{♯}\text{−}{D}_i$ axioms introduced by Vithya [12], with certiain lower separation axioms characterized through these schemes.

A continuous function $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ is termed perfect if $\mathfrak{E}$ is Hausdorff, $\Psi$ is closed, and each fiber ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is a compact subset of $\mathfrak{E}$. First introduced for metric spaces by Vainstein [13], perfect functions were independently studied by Leray [14] and Bourbaki [15] in locally compact spaces and by Yousif [16] in fibrewise ij-perfect bitopological spaces. Recent work by Atoom et al. [17,18,19,20] generalized this notion to Lindelöf perfect functions.

Throughout this paper, $(\mathfrak{E},{\rm Y})$ and $(\mathfrak{R},\Pi )$ denote arbitrary topological spaces unless specified. Product topologies are denoted as ${{\rm Y}}_1\times {{\rm Y}}_2$. Standard notations include $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{N}$, and $\mathbb{Q}$ for real numbers, integers, natural numbers, and rationals, respectively. For cardinalities, we write ${\aleph }_0$ for the cardinality of any countably infinite set and $\mathfrak{c}$ for the cardinality of the continuum, so that $\mathfrak{c}=\left|\mathbb{R}\right|=2^{{\aleph }_0}$.

This paper is organized as follows: Section 2 introduces D-Lindelöf spaces and their fundamental features, relating them to other topological spaces. Section 3 expands on this foundation by defining D-Lindelöf perfect functions and presenting examples to explain their behavior. Section 4 explores the features of these functions, their interactions with other types of spaces, and their significance in maintaining topological invariants. Section 5 investigates how D-Lindelöf perfect functions maintain essential topological features, including compactness, regularity, and paracompactness. In Section 6, we introduce three properness variants tailored to D-covers. Finally, in Section 7, we apply topological mechanisms to statistics with explicit assumptions. This systematic approach ensures a full comprehension of D-Lindelöf perfect functions, from foundational concepts to advanced applications.

2. Preliminaries

In this section, we introduce the notion of D-lindelöf spaces in topological spaces, present several of their properties, and relate them to other spaces.

We begin by recalling standard definitions that will be used in the remainder of the paper.

Definition 1 

([21]). A cover $\tilde{D}=\left\{D_{\kappa }:\kappa \in \Lambda \right\}$ of a topological space $\left(\mathfrak{E},{\rm Y}\right)$ is referred to as a D-cover if every $D_{\kappa }$ is a D-set for all $\kappa \in \Lambda$.

Definition 2 

([22]). A topological space $\left(\mathfrak{E},{\rm Y}\right)$ is termed D-lindelöf if every D-cover of $\left(\mathfrak{E},{\rm Y}\right)$ admits a countable subcover.

Definition 3 

([1]). A subset ${\mathfrak{E}}_1\subseteq (\mathfrak{E},{\rm Y})$ is called a D-set provided that there exist open sets T and A with $T\ne \mathfrak{E}$ such that ${\mathfrak{E}}_1=T\setminus A$. In this case, ${\mathfrak{E}}_1$ is said to be the D-set generated by the pair of open sets T and A.

Definition 4 

([23]). A function $\Psi :(\mathfrak{E},c)\to (\mathfrak{R},\Pi )$ is termed D-perfect if Ψ is continuous and closed, and for every $\mathfrak{r}\in \mathfrak{R}$, ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-compact.

Definition 5 

([24]). A space $\left(\mathfrak{E},{\rm Y}\right)$ is termed locally indiscrete if every open set in $\mathfrak{E}$ is open.

Definition 6 

([24]). Let $(\mathfrak{E},{\rm Y})$ be a topological space. A collection $\mathcal{L}\subseteq {\rm Y}$ is called an open cover of $\mathfrak{E}$ if

$$\mathfrak{E}=\bigcup \mathcal{L}.$$

Definition 7 

([25]). A topological space $(\mathfrak{E},{\rm Y})$ is termed D-compact if every D-cover of $\mathfrak{E}$ admits a finite subcover.

Definition 8 

([21]). A topological space $(\mathfrak{E},{\rm Y})$ is called D-countably compact if every countable D-cover of $(\mathfrak{E},{\rm Y})$ admits a finite subcover.

Definition 9 

([26]). Let $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ be a function.

(a) 

Ψ is said to be strongly continuous if

$$\Psi \left(\overline{A}\right)\subseteq \overline{\Psi \left(A\right)}\mathit{for}\mathit{every}A\subseteq \mathfrak{E},$$

where the closures are taken with respect to Υ and Π, respectively.

(b) 

Ψ is strongly continuous surjective if it is strongly continuous and surjective, that is, if it is strongly continuous and surjective, that is, 

$$\Psi \left(\mathfrak{E}\right)=\mathfrak{R}.$$

Definition 10 

([27]). Let $\mathcal{E}$ be the collection of all D-covers of a topological group $(M,\mu ,\tau )$. The infinite game $G_D\left(M\right)$ is defined as follows:

• In inning $n\in \mathbb{N}$, Player I chooses a D-cover $A_n\in \mathcal{E}$.
• Player II then chooses a nonempty finite subset $B_n\subseteq A_n$ (or a nonempty countable subset in the D-Lindelöf version).
• Player II wins the play if

  $$\bigcup _{n\in \mathbb{N}}\bigcup B_n$$

  is a D-cover of $(M,\mu ,\tau )$; otherwise, Player I wins.

Definition 11 

([28]). A Borel probability measure μ on a Hausdorff space $\mathfrak{E}$ is called a Radon probability measure if it behaves well with respect to the topology of $\mathfrak{E}$:

• When it is inner regular from inside by compact sets;
• When it is outer regular from outside by open sets.

Equivalently, μ is Radon if it is tight: for every $\epsilon >0$, there exists a compact set $K\subseteq \mathfrak{E}$ with $\mu \left(K\right)>1-\epsilon$.

Definition 12 

([29]). A class $\mathcal{F}$ is uniform Glivenko–Cantelli if ${sup}_{f\in \mathcal{F}}|P_{n}f-Pf|\to 0$ almost surely.

3. Conceptualizing $\mathit{D}$-Lindelöf Perfect Functions with Practical Examples

This section introduces the notions of D-Lindelöf perfect functions in topological spaces and provides illustrative examples to clarify the definition.

Definition 13. 

A function $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ is termed a D-Lindelöf perfect function if Ψ is continuous and closed and ${\Psi }^{-1}\left(\mathfrak{r}\right)$ for every $\mathfrak{r}\in \mathfrak{R}$, is a D-Lindelöf set in $\mathfrak{E}$.

Example 1. 

Consider $\mathfrak{E}=\mathbb{R}$ equipped with the lower limit topology ${\rm Y}=\left\{\right[a,b):a<b\}\cup \{\varnothing ,\mathbb{R}\}$. Let $\mathfrak{R}=\mathbb{R}$ with the usual topology Π, and define

$$\Psi :(\mathbb{R},{\rm Y})⟶(\mathbb{R},\Pi ),\Psi \left(x\right)=x.$$

Then, Ψ is a D-Lindelöf perfect function.

• Every open set in $\Pi$ is a union of open rays, each of which is a union of left–open intervals $[a,b)$ and thus open in ${\rm Y}$. Thus, ${\Psi }^{-1}\left(U\right)=U$ is open in ${\rm Y}$; therefore, $\Psi$ is continuous.
• Closed sets in the lower limit topology are of the form ${\displaystyle \bigcap _n[a_n,b_n)}$ and are hence Borel and therefore closed in the usual topology. Thus, $\Psi \left(F\right)=F$ is closed in $(\mathbb{R},\Pi )$, so $\Psi$ is a closed map.
• For each $\mathfrak{r}\in \mathbb{R}$,

  $${\Psi }^{-1}\left(\mathfrak{r}\right)=\left\{\mathfrak{r}\right\}.$$

  In the lower limit topology, every singleton is closed and Lindelöf, and every D-cover of a singleton trivially admits a countable subcover (indeed, the singleton itself is a D-set). Hence, each fiber is D-Lindelöf.

Since $\Psi$ is continuous, closed, and has D-Lindelöf fibers, it follows that $\Psi$ is a D-Lindelöf perfect function.

Example 2. 

Let $\mathfrak{E}=\mathbb{R}$ with the co–countable topology ${{\rm Y}}_{\mathrm{cc}}=\{\varnothing \}\cup \{U\subseteq \mathbb{R}:\mathbb{R}\setminus U\mathit{is}\mathit{countable}\}$, and let $\mathfrak{R}=\mathbb{R}$ with the usual topology Π. Define

$$\Psi :(\mathbb{R},{{\rm Y}}_{\mathrm{cc}})⟶(\mathbb{R},\Pi ),\Psi \left(x\right)=x.$$

Then, Ψ is a D–Lindelöf perfect function.

• If $V\in \Pi$ is open in the usual topology, then V is also open in the co-countable topology whenever $\mathbb{R}\setminus V$ is countable. Every usual open interval can be written as a union of such sets. Hence, ${\Psi }^{-1}\left(V\right)=V$ is open in ${{\rm Y}}_{\mathrm{cc}}$, so Ψ is continuous.
• Closed sets in $(\mathbb{R},{{\rm Y}}_{\mathrm{cc}})$ have the form $\mathbb{R}\setminus C$, where C is countable. Such sets are closed in the usual topology as countable sets have an empty interior. Thus, $\Psi \left(F\right)=F$ is closed in $(\mathbb{R},\Pi )$, so Ψ is a closed map.
• For each $\mathfrak{r}\in \mathbb{R}$,

  $${\Psi }^{-1}\left(\mathfrak{r}\right)=\left\{\mathfrak{r}\right\}.$$

  In the co-countable topology, singletons are closed and D-sets are precisely differences $U\setminus V$ with $U,V$ being co-countable. Any D-cover of a singleton contains at least one co-countable set, and thus, a countable subcover exists trivially. Therefore, each fiber is D–Lindelöf.
  Since Ψ is continuous, closed, and has D–Lindelöf fibers, it is a D–Lindelöf perfect function.

Example 3. 

Let $\mathfrak{E}=\mathbb{R}$ with the usual topology ${{\rm Y}}_l$, and $\mathfrak{R}=\mathbb{R}$ with the discrete topology ${\Pi }_d$. Define the identity function $\Psi :(\mathfrak{E},{{\rm Y}}_l)\to (\mathfrak{R},{\Pi }_d)$ by $\Psi \left(\mathfrak{e}\right)=\mathfrak{e}$. Hence, Ψ is not a D-Lindelöf perfect function.

Ψ is closed but it is not continuous. Then, Ψ is not a D-Lindelöf perfect function.

Corollary 1. 

If $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ is a D-Lindelöf perfect function, Ψ is a Lindelöf perfect function. The inverse is not true in general.

Theorem 1. 

Let $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ be a continuous and closed function. Assume that in $(\mathfrak{E},{\rm Y})$, the following equivalence holds:

$$A\subseteq \mathfrak{E}\mathit{is}\mathit{compact}⟺A\mathit{is}D\text{−}\mathit{Lindel}\mathsf{ö}\mathit{f}.$$

Then, Ψ is a perfect mapping if and only if Ψ is a D-Lindelöf perfect function.

Proof. 

Suppose first that $\Psi$ is perfect. Then, $\Psi$ is continuous and closed, and every ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is compact. Based on the hypothesis on $(\mathfrak{E},{\rm Y})$, compact subsets coincide with D-Lindelöf subsets; hence, each ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-Lindelöf. Thus, $\Psi$ is a D-Lindelöf perfect function.

Conversely, assume that $\Psi$ is a D-Lindelöf perfect function. By definition, $\Psi$ is continuous and closed, and for every $\mathfrak{r}\in \mathfrak{R}$, the ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-Lindelöf. Using again the hypothesis that D-Lindelöf subsets of $\mathfrak{E}$ are precisely the compact subsets, we conclude that every ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is compact. Therefore, $\Psi$ is a classical perfect mapping. □

Example 4. 

Let $\mathfrak{E}=\mathbb{R}$ with the usual topology and let $\mathfrak{R}={\mathbb{R}}_{\ge 0}$ with the usual topology. Define

$$\Psi :\mathbb{R}\to {\mathbb{R}}_{\ge 0},\Psi \left(x\right)=x^2.$$

(1) $\Psi$ is not a perfect function. For every $y\ge 0$, the fiber 

$${\Psi }^{-1}\left(y\right)=\left\{\begin{array}{cc}\left\{0\right\},\hfill & y=0,\hfill \\ \{-\sqrt{y},\sqrt{y}\},\hfill & y>0,\hfill \end{array}\right.$$

is finite and therefore compact. However, Ψ is not closed in the usual topology. Indeed, the closed set $(-\infty ,0]$ has

$$\Psi \left(\right(-\infty ,0\left]\right)=[0,\infty ),$$

which is not closed in ${\mathbb{R}}_{\ge 0}$. Consequently, Ψ is not a perfect function.

(2) $\Psi$ is a D–Lindelöf perfect function. Finite sets are D–Lindelöf for all difference-based structures commonly used in generalized compactness theory. Hence, every ${\Psi }^{-1}\left(y\right)$ is D–Lindelöf in $\mathfrak{E}$.

Furthermore, Ψ is D–closed: for any closed set $A\subseteq \mathbb{R}$, the image $\Psi \left(A\right)$ is an interval or a closed ray, and such sets are closed with respect to the D–topology on $\mathfrak{R}$ induced by difference sets. Since Ψ is continuous, closed in the D–sense, and has D–Lindelöf fibers, it satisfies the definition of a D–Lindelöf perfect function.

4. The Structure and Behavior of $\mathit{D}$-Lindelöf Perfect Functions

In this section, we study D-Lindelöf perfect functions in topological spaces, analyze their core properties, and relate them to other classes of spaces.

Theorem 2. 

Let $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ be a D–Lindelöf perfect function. If $Z\subseteq \mathfrak{R}$ is a D–Lindelöf subspace, then ${\Psi }^{-1}\left(Z\right)$ is D–Lindelöf in $(\mathfrak{E},{\rm Y})$.

Proof. 

Let $\tilde{D}=\{D_{\kappa }:\kappa \in \Lambda \}$ be a D–cover of ${\Psi }^{-1}\left(Z\right)$, that is,

$${\Psi }^{-1}\left(Z\right)\subseteq \bigcup _{\kappa \in \Lambda }{D}_{\kappa }\mathrm{and}\mathrm{each}{D}_{\kappa }\mathrm{is}\mathrm{a}D–\mathrm{set}\mathrm{in}(\mathfrak{E},{\rm Y}).$$

For each $\mathfrak{r}\in Z$, the fiber ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D–Lindelöf (because $\Psi$ is D–Lindelöf perfect). Hence, there exists a countable index set ${\Lambda }_{\mathfrak{r}}\subseteq \Lambda$ such that

$${\Psi }^{-1}\left(\mathfrak{r}\right)\subseteq \bigcup _{\kappa \in {\Lambda }_{\mathfrak{r}}}{D}_{\kappa }.$$

Set

$$U_{\mathfrak{r}}:=\bigcup _{\kappa \in {\Lambda }_{\mathfrak{r}}}{D}_{\kappa }.$$

Since $\Psi$ is closed, the set $\Psi (\mathfrak{E}\setminus U_{\mathfrak{r}})$ is closed in $(\mathfrak{R},\Pi )$; hence,

$$O_{\mathfrak{r}}:=\mathfrak{R}\setminus \Psi (\mathfrak{E}\setminus U_{\mathfrak{r}})$$

is open in $(\mathfrak{R},\Pi )$. We now show the following:

(1)

$\mathfrak{r}\in O_{\mathfrak{r}}$. If $\mathfrak{r}\notin O_{\mathfrak{r}}$, then $\mathfrak{r}\in \Psi (\mathfrak{E}\setminus U_{\mathfrak{r}})$, so there exists $x\in \mathfrak{E}\setminus U_{\mathfrak{r}}$ with $\Psi \left(x\right)=\mathfrak{r}$, i.e., $x\in {\Psi }^{-1}\left(\mathfrak{r}\right)\setminus U_{\mathfrak{r}}$, contradicting ${\Psi }^{-1}\left(\mathfrak{r}\right)\subseteq U_{\mathfrak{r}}$.

(2)

${\Psi }^{-1}\left(O_{\mathfrak{r}}\right)\subseteq U_{\mathfrak{r}}$. If $x\in {\Psi }^{-1}\left(O_{\mathfrak{r}}\right)$, then $\Psi \left(x\right)\notin \Psi (\mathfrak{E}\setminus U_{\mathfrak{r}})$, and hence, $x\notin \mathfrak{E}\setminus U_{\mathfrak{r}}$; thus, $x\in U_{\mathfrak{r}}$.

Now consider the subspace $(Z,{\Pi }_Z)$, where ${\Pi }_Z$ is the topology induced from $\Pi$. The family

$$\tilde{O}:=\{O_{\mathfrak{r}}\cap Z:\mathfrak{r}\in Z\}$$

is an open cover of Z (because $\mathfrak{r}\in O_{\mathfrak{r}}\cap Z$ for each $\mathfrak{r}\in Z$). By our convention that every open set is a D–set (since $U=U\setminus ⌀$), $\tilde{O}$ is a D–cover of Z.

Because Z is D–Lindelöf, there exists a countable subfamily $\{O_{{\mathfrak{r}}_i}\cap Z:i\in \mathbb{N}\}\subseteq \tilde{O}$ with

$$Z\subseteq \bigcup _{i=1}^{\infty }(O_{{\mathfrak{r}}_i}\cap Z).$$

Applying ${\Psi }^{-1}$, we obtain

$${\Psi }^{-1}\left(Z\right)\subseteq \bigcup _{i=1}^{\infty }{\Psi }^{-1}\left(O_{{\mathfrak{r}}_i}\right)\subseteq \bigcup _{i=1}^{\infty }{U}_{{\mathfrak{r}}_i}=\bigcup _{i=1}^{\infty }\bigcup _{\kappa \in {\Lambda }_{{\mathfrak{r}}_i}}{D}_{\kappa }.$$

The index set

$${\Lambda }^{*}:=\bigcup _{i=1}^{\infty }{\Lambda }_{{\mathfrak{r}}_i}$$

is a countable union of countable sets and thus countable. Therefore, $\{D_{\kappa }:\kappa \in {\Lambda }^{*}\}$ is a countable subfamily of $\tilde{D}$ that covers ${\Psi }^{-1}\left(Z\right)$. Thus, ${\Psi }^{-1}\left(Z\right)$ is D–Lindelöf in $(\mathfrak{E},{\rm Y})$. □

Corollary 2. 

A D-Lindelöf space is preserved under preimages of D-Lindelöf perfect functions.

Theorem 3. 

If $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\varsigma )$ and $\rho :(\mathfrak{R},\varsigma )\to (Z,\Pi )$ are D-Lindelöf perfect functions, then $\rho \circ \Psi :(\mathfrak{E},{\rm Y})\to (Z,\Pi )$ is also a D-Lindelöf perfect function.

Proof. 

The composition $\rho \circ \Psi$ is continuous, as both $\Psi$ and $\rho$ are continuous by definition. Let $C\subseteq Z$ be closed. Then,

$${(\rho \circ \Psi )}^{-1}\left(C\right)={\Psi }^{-1}\left({\rho }^{-1}\left(C\right)\right).$$

Since $\rho$ is closed, ${\rho }^{-1}\left(C\right)$ is closed in $\mathfrak{R}$. As $\Psi$ is closed, ${\Psi }^{-1}\left({\rho }^{-1}\left(C\right)\right)$ is closed in $\mathfrak{E}$. Thus, $\rho \circ \Psi$ preserves closedness.

For any $z\in Z$,

$${(\rho \circ \Psi )}^{-1}\left(z\right)={\Psi }^{-1}\left({\rho }^{-1}\left(z\right)\right).$$

Since $\rho$ is D-Lindelöf perfect, ${\rho }^{-1}\left(z\right)$ is a D-Lindelöf subset of $\mathfrak{R}$. By Theorem 2, ${\Psi }^{-1}\left({\rho }^{-1}\left(z\right)\right)$ is D-Lindelöf in $\mathfrak{E}$.

Thus, $\rho \circ \Psi$ satisfies all conditions of a D-Lindelöf perfect function. □

Theorem 4. 

Let

$$\Psi :(\mathfrak{E},{\rm Y})⟶(\mathfrak{R},\Pi )$$

be a homeomorphism and let

$$\rho :(\mathfrak{R},\Pi )\stackrel{onto}{⟶}(Z,\Theta )$$

be a continuous function. If the composition

$$\rho \circ \Psi :(\mathfrak{E},{\rm Y})⟶(Z,\Theta )$$

is a D-Lindelöf perfect function, then ρ is also a D-Lindelöf perfect function.

Proof. 

Since $\Psi$ is a homeomorphism, both $\Psi$ and ${\Psi }^{-1}$ are continuous and closed, and $\Psi$ is bijective.

Write

$$\rho =(\rho \circ \Psi )\circ {\Psi }^{-1}.$$

The map $\rho \circ \Psi$ is D-Lindelöf perfect by assumption, hence closed. The inverse ${\Psi }^{-1}$ is a homeomorphism, so it is also closed. The composition of two closed maps is closed, hence $\rho$ is closed.

Let $z\in Z$. Since $\rho \circ \Psi$ is D-Lindelöf perfect, its fibre

$${(\rho \circ \Psi )}^{-1}\left(z\right)\subseteq \mathfrak{E}$$

is D-Lindelöf in $(\mathfrak{E},{\rm Y})$ by definition.

Because $\Psi$ is bijective, we have

$${\rho }^{-1}\left(z\right)=\{\mathfrak{r}\in \mathfrak{R}:\rho \left(\mathfrak{r}\right)=z\}=\Psi \left({(\rho \circ \Psi )}^{-1}\left(z\right)\right).$$

Let $Y\subseteq \mathfrak{E}$ be D-Lindelöf in $(\mathfrak{E},{\rm Y})$ and put $\widehat{Y}:=\Psi \left(Y\right)\subseteq \mathfrak{R}$. Let $\mathcal{L}=\{L_{\alpha }:\alpha \in A\}$ be a D-cover of $\widehat{Y}$ in $(\mathfrak{R},\Pi )$, that is, each $L_{\alpha }$ is a D-set in $(\mathfrak{R},\Pi )$ and $\widehat{Y}\subseteq {\bigcup }_{\alpha \in A}{L}_{\alpha }$.

For every $\alpha$,

$$L_{\alpha }^{\prime }:={\Psi }^{-1}\left(L_{\alpha }\right)$$

is a D-set in $(\mathfrak{E},{\rm Y})$. The family ${\mathcal{L}}^{\prime }:=\{L_{\alpha }^{\prime }:\alpha \in A\}$ is a D-cover of Y, because

$$Y={\Psi }^{-1}\left(\widehat{Y}\right)\subseteq {\Psi }^{-1}\left(\bigcup _{\alpha \in A}{L}_{\alpha }\right)=\bigcup _{\alpha \in A}{L}_{\alpha }^{\prime }.$$

Since Y is D-Lindelöf, there exists a countable index set $A_0\subseteq A$ such that

$$Y\subseteq \bigcup _{\alpha \in A_0}{L}_{\alpha }^{\prime }.$$

Applying $\Psi$ gives

$$\widehat{Y}=\Psi \left(Y\right)\subseteq \bigcup _{\alpha \in A_0}\Psi \left(L_{\alpha }^{\prime }\right)=\bigcup _{\alpha \in A_0}{L}_{\alpha }.$$

Thus, every D-cover of $\widehat{Y}$ in $(\mathfrak{R},\Pi )$ admits a countable subcover, and $\widehat{Y}$ is D-Lindelöf.

Applying this to $Y={(\rho \circ \Psi )}^{-1}\left(z\right)$ and $\widehat{Y}={\rho }^{-1}\left(z\right)$ shows that ${\rho }^{-1}\left(z\right)$ is D-Lindelöf for every $z\in Z$.

Hence $\rho$ is a D-Lindelöf perfect function. □

Theorem 5. 

Let there be a topological space $(\mathfrak{E},{\rm Y})$ and $B\subseteq \mathfrak{E}$. The subspace $(B,{{\rm Y}}_B)$ is D-Lindelöf if and only if every cover of B through D-sets in $\mathfrak{E}$ admits countable subcover.

Proof. 

(⇒) Assume $(B,{{\rm Y}}_B)$ is D-Lindelöf. Let $\tilde{\mathcal{D}}=\{D_{\kappa }:\kappa \in \Lambda \}$ be a cover of B by D-sets in $\mathfrak{E}$, where every $D_{\kappa }=L_{\kappa }\setminus T_{\kappa }$ for open sets $L_{\kappa },T_{\kappa }\in {\rm Y}$ with $L_{\kappa }\ne \mathfrak{E}$. In ${{\rm Y}}_B$, $D_{\kappa }\cap B$ is a D-open set in B. Thus, $\{D_{\kappa }\cap B:\kappa \in \Lambda \}$ is a D-open cover of B. Then, there exists ${\{D_{{\kappa }_i}\cap B\}}_{i=1}^{\infty }$ that covers B. Consequently, ${\left\{D_{{\kappa }_i}\right\}}_{i=1}^{\infty }$ is a countable subcover of B in $\mathfrak{E}$.

(⇐) Assume each cover of B by D-sets in $\mathfrak{E}$ admits a countable subcover. Consider $\tilde{\mathcal{L}}=\{L_{\beta }:\beta \in \Gamma \}$ is a D-open cover of B in ${{\rm Y}}_B$. Therefore, $\{D_{\beta }:\beta \in \Gamma \}$ covers B in $\mathfrak{E}$. Then, ${\{D_{{\beta }_j}\cap B\}}_{j=1}^{\infty }$ is a countable subcover of $\tilde{\mathcal{L}}$ in ${{\rm Y}}_B$. Hence, $(B,{{\rm Y}}_B)$ is D-Lindelöf. □

Theorem 6. 

Let $\Psi :(\mathfrak{E},{\rm Y})\stackrel{onto}{\to }(\mathfrak{R},\Pi )$ be a D-Lindelöf perfect function. For every subset $B\subset \mathfrak{R}$, ${\Psi }_B:{\Psi }^{-1}\left(B\right)\to B$ is also D-Lindelöf perfect.

Proof. 

Let $C\subseteq {\Psi }^{-1}\left(B\right)$ be closed. Then, $C=D\cap {\Psi }^{-1}\left(B\right)$ for some closed $D\subseteq \mathfrak{E}$. Since $\Psi$ is closed, $\Psi \left(D\right)$ is closed in $\mathfrak{R}$, thus

$${\Psi }_B\left(C\right)=\Psi \left(D\right)\cap B$$

is closed in B. Hence, ${\Psi }_B$ is closed.

The restriction ${\Psi }_B$ inherits continuity from $\Psi$. Now, for every $b\in B$, ${\Psi }_B^{-1}\left(b\right)={\Psi }^{-1}\left(b\right)\cap {\Psi }^{-1}\left(B\right)={\Psi }^{-1}\left(b\right)$. Since $\Psi$ is a D-Lindelöf perfect function, ${\Psi }^{-1}\left(b\right)$ is D-Lindelöf in $\mathfrak{E}$. Any D-cover of ${\Psi }^{-1}\left(b\right)$ in ${\Psi }^{-1}\left(B\right)$ corresponds to a D-cover in $\mathfrak{E}$, which admits a countable subcover. Thus, ${\Psi }^{-1}\left(b\right)$ remains D-Lindelöf in ${\Psi }^{-1}\left(B\right)$.

Since ${\Psi }_B$ is closed and continuous and has D-Lindelöf fibers, it is D-Lindelöf perfect. □

Theorem 7. 

Let there be a continuous bijection function $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$. If $(\mathfrak{R},\Pi )$ is a Hausdorff space and $(\mathfrak{E},{\rm Y})$ is D-Lindelöf, it follows that Ψ is a homeomorphism.

Proof. 

Since $(\mathfrak{E},{\rm Y})$ is D-Lindelöf and closed subsets of D-Lindelöf spaces are D-Lindelöf, F is D-Lindelöf. The continuous image of a D-Lindelöf set is D-Lindelöf. Thus, $\Psi \left(F\right)$ is D-Lindelöf in $\mathfrak{R}$.

D-Lindelöf subsets of a Hausdorff space are closed. Hence, $\Psi \left(F\right)$ is closed in $\mathfrak{R}$.

Since $\Psi$ is a continuous, closed bijection, it is a homeomorphism. □

Theorem 8. 

Let $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ be a strongly continuous surjective function. If $(\mathfrak{R},\Pi )$ is D–Lindelöf, then $(\mathfrak{E},{\rm Y})$ is D–Lindelöf.

Proof. 

Let $\tilde{\mathcal{L}}=\{L_{\kappa }:\kappa \in \Lambda \}$ be a D–cover of $\mathfrak{E}$. For each $L_{\kappa }$, write $L_{\kappa }=U_{\kappa }\setminus V_{\kappa }$ where $U_{\kappa },V_{\kappa }$ are open in $(\mathfrak{E},{\rm Y})$.

Strong continuity means

$$\Psi \left(\overline{A}\right)\subseteq \overline{\Psi \left(A\right)}\mathrm{for}\mathrm{every}A\subseteq \mathfrak{E}.$$

For each $\kappa \in \Lambda$, define

$$T_{\kappa }:=\mathfrak{R}\setminus \Psi (\mathfrak{E}\setminus L_{\kappa }).$$

Since $\Psi (\mathfrak{E}\setminus L_{\kappa })$ is closed (strong continuity implies images of closed sets are contained in closed sets), $T_{\kappa }$ is open in $\mathfrak{R}$ and hence a D–set (open $=O\setminus ⌀$).

We claim that the family $\{T_{\kappa }:\kappa \in \Lambda \}$ is a D–cover of $\mathfrak{R}$.

Let $\mathfrak{r}\in \mathfrak{R}$. Since $\Psi$ is surjective, choose $x\in \mathfrak{E}$ with $\Psi \left(x\right)=\mathfrak{r}$. As $\tilde{\mathcal{L}}$ covers $\mathfrak{E}$, there exists $\kappa$ with $x\in L_{\kappa }$. Then, $x\notin \mathfrak{E}\setminus L_{\kappa }$, so $\Psi \left(x\right)=\mathfrak{r}\notin \Psi (\mathfrak{E}\setminus L_{\kappa })$, hence $\mathfrak{r}\in T_{\kappa }$. Thus,

$$\mathfrak{R}=\bigcup _{\kappa \in \Lambda }{T}_{\kappa },$$

so $\left\{T_{\kappa }\right\}$ is a D–cover of $\mathfrak{R}$.

Since $(\mathfrak{R},\Pi )$ is D–Lindelöf, there exists a countable index set ${\Lambda }^{*}\subseteq \Lambda$ such that

$$\mathfrak{R}\subseteq \bigcup _{\kappa \in {\Lambda }^{*}}{T}_{\kappa }.$$

For each $\kappa \in \Lambda$, we have

$${\Psi }^{-1}\left(T_{\kappa }\right)={\Psi }^{-1}\left(\mathfrak{R}\setminus \Psi (\mathfrak{E}\setminus L_{\kappa })\right)=\mathfrak{E}\setminus {\Psi }^{-1}\left(\Psi (\mathfrak{E}\setminus L_{\kappa })\right)\subseteq \mathfrak{E}\setminus (\mathfrak{E}\setminus L_{\kappa })=L_{\kappa }.$$

Thus

$${\Psi }^{-1}\left(T_{\kappa }\right)\subseteq L_{\kappa }.$$

Using surjectivity,

$$\mathfrak{E}={\Psi }^{-1}\left(\mathfrak{R}\right)\subseteq {\Psi }^{-1}\left(\bigcup _{\kappa \in {\Lambda }^{*}}{T}_{\kappa }\right)=\bigcup _{\kappa \in {\Lambda }^{*}}{\Psi }^{-1}\left(T_{\kappa }\right)\subseteq \bigcup _{\kappa \in {\Lambda }^{*}}{L}_{\kappa }.$$

Therefore, $\{L_{\kappa }:\kappa \in {\Lambda }^{*}\}$ is a countable D–subcover of $\mathfrak{E}$. Hence, $(\mathfrak{E},{\rm Y})$ is D–Lindelöf. □

5. Preservation of Topological Properties via $\mathit{D}$-Lindelöf Perfect Functions

This section explores how D-Lindelöf perfect functions preserve fundamental topological properties across spaces.

Theorem 9. 

Let $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ be a D-Lindelöf perfect function, that is, Ψ is continuous and closed, and each fiber ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-Lindelöf. Assume in addition that for every $\mathfrak{r}\in \mathfrak{R}$, ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-countably compact. If $(\mathfrak{R},\Pi )$ is D-countably compact, then $(\mathfrak{E},{\rm Y})$ is D-countably compact.

Proof. 

Let $\{L_n:n\in \mathbb{N}\}$ be a countable D-cover of $(\mathfrak{E},{\rm Y})$, that is, each $L_n$ is a D-set in $(\mathfrak{E},{\rm Y})$ and

$$\mathfrak{E}\subseteq \bigcup _{n\in \mathbb{N}}{L}_n.$$

For each n, write $L_n=U_n\setminus V_n$ with $U_n,V_n$ open in $(\mathfrak{E},{\rm Y})$. Define

$$O_n:=\mathfrak{R}\setminus \Psi \left(\mathfrak{E}\setminus L_n\right)(n\in \mathbb{N}).$$

Since $\mathfrak{E}\setminus L_n$ is closed and $\Psi$ is closed, $\Psi (\mathfrak{E}\setminus L_n)$ is closed in $(\mathfrak{R},\Pi )$; hence, each $O_n$ is open in $\mathfrak{R}$, and therefore a D-set.

Claim 1. 

The family $\{O_n:n\in \mathbb{N}\}$ is a D-cover of $(\mathfrak{R},\Pi )$.

Let $\mathfrak{r}\in \mathfrak{R}$. Then the fiber ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is nonempty (by surjectivity of $\Psi$) and D-countably compact by hypothesis. The family

$${\mathcal{L}}_{\mathfrak{r}}:=\{L_n\cap {\Psi }^{-1}\left(\mathfrak{r}\right):n\in \mathbb{N}\}$$

is a countable D-cover of the subspace ${\Psi }^{-1}\left(\mathfrak{r}\right)$, since $\left\{L_n\right\}$ covers $\mathfrak{E}$ and intersections of D-sets with a subspace are D-sets in that subspace.

Because ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-countably compact, there exists a finite set $F_{\mathfrak{r}}\subseteq \mathbb{N}$ such that

$${\Psi }^{-1}\left(\mathfrak{r}\right)\subseteq \bigcup _{n\in F_{\mathfrak{r}}}\left(L_n\cap {\Psi }^{-1}\left(\mathfrak{r}\right)\right)=\bigcup _{n\in F_{\mathfrak{r}}}{L}_n.$$

Suppose, toward a contradiction, that $\mathfrak{r}\notin O_n$ for every $n\in F_{\mathfrak{r}}$. Then for each such n, $\mathfrak{r}\in \Psi (\mathfrak{E}\setminus L_n)$, so there exists $x_n\in \mathfrak{E}\setminus L_n$ with $\Psi \left(x_n\right)=\mathfrak{r}$. Thus $x_n\in {\Psi }^{-1}\left(\mathfrak{r}\right)\setminus {\bigcup }_{n\in F_{\mathfrak{r}}}{L}_n$, contradicting the above inclusion ${\Psi }^{-1}\left(\mathfrak{r}\right)\subseteq {\bigcup }_{n\in F_{\mathfrak{r}}}{L}_n$.

Hence, there exists some $n\in F_{\mathfrak{r}}$ with $\mathfrak{r}\in O_n$. Therefore, 

$$\mathfrak{R}=\bigcup _{\mathfrak{r}\in \mathfrak{R}}\left\{\mathfrak{r}\right\}\subseteq \bigcup _{n\in \mathbb{N}}{O}_n,$$

and $\left\{O_n\right\}$ is a countable D-cover of $(\mathfrak{R},\Pi )$.

Claim 2. 

$(\mathfrak{E},{\rm Y})$ is D-countably compact.

Since $(\mathfrak{R},\Pi )$ is D-countably compact and $\{O_n:n\in \mathbb{N}\}$ is a countable D-cover of $\mathfrak{R}$, there exists a finite set $F\subseteq \mathbb{N}$ such that 

$$\mathfrak{R}\subseteq \bigcup _{n\in F}{O}_n.$$

Then

$$\mathfrak{E}={\Psi }^{-1}\left(\mathfrak{R}\right)\subseteq {\Psi }^{-1}\left(\bigcup _{n\in F}{O}_n\right)=\bigcup _{n\in F}{\Psi }^{-1}\left(O_n\right).$$

By construction, 

$${\Psi }^{-1}\left(O_n\right)={\Psi }^{-1}\left(\mathfrak{R}\setminus \Psi (\mathfrak{E}\setminus L_n)\right)=\mathfrak{E}\setminus {\Psi }^{-1}\left(\Psi (\mathfrak{E}\setminus L_n)\right)\subseteq \mathfrak{E}\setminus (\mathfrak{E}\setminus L_n)=L_n,$$

so ${\Psi }^{-1}\left(O_n\right)\subseteq L_n$ for each n. Hence,

$$\mathfrak{E}\subseteq \bigcup _{n\in F}{L}_n.$$

We have shown that every countable D-cover $\left\{L_n\right\}$ of $\mathfrak{E}$ admits a finite subcover. Therefore, $(\mathfrak{E},{\rm Y})$ is D-countably compact. □

Theorem 10. 

If $(\mathfrak{E},{\rm Y})$ is normal Hausdorff, and $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ is a surjective D-Lindelöf perfect function with closed fibers, then $(\mathfrak{R},\Pi )$ is Hausdorff.

Proof. 

Let $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ be a surjective D-Lindelöf perfect function with closed fibers, where $(\mathfrak{E},{\rm Y})$ is normal Hausdorff. We show that $(\mathfrak{R},\Pi )$ is Hausdorff.

Let ${\mathfrak{r}}_1\ne {\mathfrak{r}}_2$ in $\mathfrak{R}$. Since $\Psi$ is D-Lindelöf perfect, ${\Psi }^{-1}\left({\mathfrak{r}}_1\right)$ and ${\Psi }^{-1}\left({\mathfrak{r}}_2\right)$ are D-Lindelöf, fibers are closed, and ${\Psi }^{-1}\left({\mathfrak{r}}_1\right)\cap {\Psi }^{-1}\left({\mathfrak{r}}_2\right)=\varnothing$.

As $(\mathfrak{E},{\rm Y})$ is normal, disjoint closed sets can be separated. Thus, ∃ disjoint open sets $L,T\subseteq \mathfrak{E}$:

$${\Psi }^{-1}\left({\mathfrak{r}}_1\right)\subseteq L,{\Psi }^{-1}\left({\mathfrak{r}}_2\right)\subseteq T,L\cap T=\varnothing .$$

Since $\Psi$ is D-Lindelöf perfect which implies that it is closed, $\Psi (\mathfrak{E}\setminus L)$ and $\Psi (\mathfrak{E}\setminus T)$ are closed in $\mathfrak{R}$. Define

$$W_1=\mathfrak{R}\setminus \Psi (\mathfrak{E}\setminus L),W_2=\mathfrak{R}\setminus \Psi (\mathfrak{E}\setminus T).$$

$W_1$ and $W_2$ are open in $\mathfrak{R}$. Now, ${\mathfrak{r}}_1\in W_1$: If ${\mathfrak{r}}_1\in \Psi (\mathfrak{E}\setminus L)$, then $\exists z\in \mathfrak{E}\setminus L$ with $\Psi \left(z\right)={\mathfrak{r}}_1$, so $z\in {\Psi }^{-1}\left({\mathfrak{r}}_1\right)\setminus L$, representing a contradiction. Thus, ${\mathfrak{r}}_1\in W_1$. Similarly, ${\mathfrak{r}}_2\in W_2$.

Finally,

$$W_1\cap W_2=\mathfrak{R}\setminus \left(\Psi (\mathfrak{E}\setminus L)\cup \Psi (\mathfrak{E}\setminus T)\right)$$

Since $L\cap T=\varnothing$, we have $(\mathfrak{E}\setminus L)\cup (\mathfrak{E}\setminus T)=\mathfrak{E}$, so

$$W_1\cap W_2=\mathfrak{R}\setminus \Psi \left(\mathfrak{E}\right)=\mathfrak{R}\setminus \mathfrak{R}=\varnothing (\mathrm{by}\mathrm{surjectivity}).$$

Thus, $W_1$ and $W_2$ are disjoint open neighborhoods separating ${\mathfrak{r}}_1$ and ${\mathfrak{r}}_2$. □

Theorem 11. 

Let $(\mathfrak{E},{\rm Y})$ and $(\mathfrak{R},\Pi )$ be topological spaces. If $(\mathfrak{E},{\rm Y})$ is D-Lindelöf and $(\mathfrak{R},\Pi )$ is D-compact, then $\pi :(\mathfrak{E}\times \mathfrak{R},{\rm Y}\times \Pi )\to (\mathfrak{R},\Pi )$ is closed.

Proof. 

Since $(\mathfrak{E},{\rm Y})$ is a D-Lindelöf and $(\mathfrak{R},\Pi )$ is D-compact, then $(\mathfrak{E}\times \mathfrak{R},{\rm Y}\times \Pi )$ is D-Lindelöf. Therefore, the projection $\pi :(\mathfrak{E}\times \mathfrak{R},{\rm Y}\times \Pi )\to (\mathfrak{R},\Pi )$ is a closed function. □

6. New Relations, Variants, and Tools for $\mathit{D}$-Lindelöf Perfect Maps

In this section, we provide three D-cover-specific properness variations, relate them to choosing rules and a point-D-open competition, and obtain stability findings (products, sums, and images) as well as sharp cardinal bounds. Stronger D-Lindelöf ideal assumptions are indicated where they are permitted, but many transitions really just require surjective continuity.

Definition 14. 

A function $\Psi :(\mathfrak{E},{\rm Y})\to (\mathfrak{R},\Pi )$ is

1. 

D-proper if for every compact $K\subseteq \mathfrak{R}$, ${\Psi }^{-1}\left(K\right)$ is D-compact in $\mathfrak{E}$;

2. 

D-quasi-perfect if Ψ is closed and every ${\Psi }^{-1}\left(\mathfrak{r}\right)$ is D-compact;

3. 

D-Lindelöf proper if Ψ is closed and ${\Psi }^{-1}\left(K\right)$ is D-Lindelöf for every compact $K\subseteq \mathfrak{R}$.

Example 5. 

Let S be an uncountable set and fix $p\in S$. Equip $\mathfrak{R}:=S$ with the Fortissimo topology Π: a set $L\subseteq S$ is open if either $L=⌀$ or $p\in L$ and $S\setminus L$ are countable. Then, every compact subset of $(\mathfrak{R},\Pi )$ is finite: if $K\subseteq S$ is infinite with $p\notin K$, the family $\{S\setminus \{\mathfrak{e}\}\cup \{p\}:\mathfrak{e}\in K\}$ is an open cover of K with no finite subcover. If K is infinite and $p\in K$, enumerate $K\setminus \left\{p\right\}=\{{\mathfrak{e}}_n:n\in \mathbb{N}\}$ and cover K by $L_n:=S\setminus \left\{{\mathfrak{e}}_n\right\}\cup \left\{p\right\}$; once more, there is no finite subcover.

Let $\mathfrak{E}$ be any infinite set with the discrete topology Υ, and select an injection $\Psi :\mathfrak{E}\to \mathfrak{R}$ whose range $B=\Psi \left(\mathfrak{E}\right)$ is infinite and $p\notin B$. Next, Ψ is continuous. Each and every compact $K\subseteq \mathfrak{R}$ we have is K finite and hence ${\Psi }^{-1}\left(K\right)$ finite. On a discrete space, D-sets correspond with open sets, so D-compactness corresponds with compactness; finite sets are D-compact. Thus, Ψ is D-proper.

Ψ is not closed: grab the closed set $A\subseteq \mathfrak{E}$ with $\Psi \left(A\right)=B$. In $(\mathfrak{R},\Pi )$, the closure of B is $B\cup \left\{p\right\}$, so B is not closed. Hence, Ψ unable to meet the closedness requirement in D-quasi-perfect.

Consequently, Ψ is D-proper but not D-quasi-perfect (and a fortiori, not D-Lindelöf proper).

Example 6. 

Let $\mathfrak{E}=\mathbb{N}\times \mathbb{N}$ with the discrete topology Υ, let $\mathfrak{R}:=\mathbb{N}$ with the discrete topology Π, and define $\Psi :\mathfrak{E}\to \mathfrak{R}$ by $\Psi (n,m)=n$.

Ψ is closed. In discrete spaces, D-sets are exactly the open sets, so D-Lindelöf ⇔ Lindelöf and D-compact ⇔ compact. Compact subsets of $(\mathfrak{R},\Pi )$ are precisely the finite sets $K\subseteq \mathbb{N}$. For such K,

$${\Psi }^{-1}\left(K\right)=K\times \mathbb{N},$$

which is countable discrete. Hence, ${\Psi }^{-1}\left(K\right)$ is Lindelöf (thus D-Lindelöf) but not compact (thus not D-compact).

As a consequence, Ψ is D-Lindelöf proper. Ψ is not D-proper, since ${\Psi }^{-1}\left(K\right)=K\times \mathbb{N}$ fails D-compactness whenever $K\ne ⌀$. Ψ is not D-quasi-perfect either, because ${\Psi }^{-1}\left(n\right)=\left\{n\right\}\times \mathbb{N}$ is not D-compact.

Proposition 1. 

For Hausdorff $\mathfrak{R}$: $D\text{−}\mathit{quasi}\text{−}\mathit{perfect}\Rightarrow D\text{−}\mathit{proper}\Rightarrow D\text{−}\mathit{Lindel}\mathsf{ö}f\mathit{proper}$. If Ψ is D-Lindelöf perfect and $\mathfrak{R}$ is σ-compact, then Ψ is D-Lindelöf proper.

Definition 15. 

${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$: for every sequence $\left({\mathcal{L}}_n\right)$ in $\mathcal{D}\left(\mathfrak{E}\right)$, choose $L_n\in {\mathcal{L}}_n$ so that $\{L_n:n\in \mathbb{N}\}$ D-covers $\mathfrak{E}$. In the same way, ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$ with finite selections.

Example 7. 

Let $\mathfrak{E}$ be countable with the discrete topology. Then every D-set is open. In the same way, every D-cover is an open cover. List $\mathfrak{E}=\{{\mathfrak{e}}_1,{\mathfrak{e}}_2,\dots \}$. Considering any sequence ${\left({\mathcal{L}}_n\right)}_{n\in \mathbb{N}}$ of D-covers of $\mathfrak{E}$, for each n, choose $L_n\in {\mathcal{L}}_n$ with ${\mathfrak{e}}_n\in L_n$ (feasible because ${\mathcal{L}}_n$ covers $\mathfrak{E}$). Then $\{L_n:n\in \mathbb{N}\}$ covers $\mathfrak{E}$, so $\mathfrak{E}$ satisfies ${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$. Consequently, ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$ holds true as well.

Example 8. 

Let $\mathfrak{E}$ be an uncountable set with the co-countable topology. Every nonempty open set is co-countable. For D-sets, if $L,T$ are open and nonempty, then $L\setminus T$ is countable; if $T=⌀$, then $L\setminus T=L$ is co-countable. Hence, any D-cover $\mathcal{L}$ of $\mathfrak{E}$ has to have a minimum of one co-countable member. Without a union of countably, many countable sets could not cover uncountable $\mathfrak{E}$.

Create a predetermined sequence $\left({\mathcal{L}}_n\right)$ where each ${\mathcal{L}}_n=\{\mathfrak{E}\setminus C:C\subset \mathfrak{E}\mathit{countable}\}\cup \{C:C\subset \mathfrak{E}\mathit{countable}\}$. Each ${\mathcal{L}}_n$ is a D-cover of $\mathfrak{E}$.

(1) 

${\mathrm{S}}_1$ is unsuccessful: Select any $L_n\in {\mathcal{L}}_n$. If infinitely many choices are countable, their union is countable; if cofinitely, many are co-countable. The union of the selected sets nevertheless loses points since the cross-section of their complements is limitless. Therefore, not just one choice $\left\{L_n\right\}$ can cover $\mathfrak{E}$.

(2) 

${\mathrm{S}}_{fin}$ fails: Choose any finite ${\mathcal{T}}_n\subseteq {\mathcal{L}}_n$. Then, ${\bigcup }_n\bigcup {\mathcal{T}}_n$ consists of a countable mixture of sets that are all countable or co-countable. The portion they cover is countable if an unlimited number of chosen sets are countable. The complements of finitely many co-countable sets have infinite intersection if cofinitely many are co-countable, leaving uncovered points. Therefore, no series of limited choices encompasses $\mathfrak{E}$.

Therefore, $\mathfrak{E}$ satisfies neither ${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$ nor ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$.

Theorem 12. 

Let $\Psi :\mathfrak{E}\to \mathfrak{R}$ be D-Lindelöf perfect and onto.

1. 

If $\mathfrak{E}$ satisfies ${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$, then $\mathfrak{R}$ satisfies ${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$.

2. 

If $\mathfrak{E}$ satisfies ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$, then $\mathfrak{R}$ satisfies ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$.

Proof. 

Let $\Psi :\mathfrak{E}\to \mathfrak{R}$ be onto and continuous.

(1)

Assume $\mathfrak{E}$ satisfies ${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$. Let $\left({\mathcal{L}}_n\right)$ be a sequence of D-covers of $\mathfrak{R}$. For each n, the pullback

$${\Psi }^{-1}\left[{\mathcal{L}}_n\right]:=\{{\Psi }^{-1}\left(L\right):L\in {\mathcal{L}}_n\}$$

is a D-cover of $\mathfrak{E}$ since $\Psi$ is onto and ${\Psi }^{-1}(L_1\setminus L_2)={\Psi }^{-1}\left(L_1\right)\setminus {\Psi }^{-1}\left(L_2\right)$ with ${\Psi }^{-1}\left(L_i\right)$ open in $\mathfrak{E}$. By ${\mathrm{S}}_1$ on $\mathfrak{E}$, pick $T_n\in {\Psi }^{-1}\left[{\mathcal{L}}_n\right]$ so that $\{T_n:n\in \mathbb{N}\}$ D-covers $\mathfrak{E}$. Write $T_n={\Psi }^{-1}\left(L_n\right)$ with $L_n\in {\mathcal{L}}_n$. Then,

$$\mathfrak{R}=\Psi \left(\mathfrak{E}\right)\subseteq \Psi \left(\bigcup _n{\Psi }^{-1}\left(L_n\right)\right)\subseteq \bigcup _n\Psi \left({\Psi }^{-1}\left(L_n\right)\right)\subseteq \bigcup _n{L}_n,$$

so $\{L_n:n\in \mathbb{N}\}$ D-covers $\mathfrak{R}$. Hence, $\mathfrak{R}$ satisfies ${\mathrm{S}}_1(\mathcal{D},\mathcal{D})$.

(2)

Assume $\mathfrak{E}$ satisfies ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$. For each n, apply ${\mathrm{S}}_{\mathrm{fin}}$ to the D-cover ${\Psi }^{-1}\left[{\mathcal{L}}_n\right]$ to get a finite ${\mathcal{T}}_n\subseteq {\Psi }^{-1}\left[{\mathcal{L}}_n\right]$ such that ${\bigcup }_n\bigcup {\mathcal{T}}_n$ D-covers $\mathfrak{E}$. Write ${\mathcal{T}}_n=\{{\Psi }^{-1}\left(L\right):L\in {\mathcal{W}}_n\}$ with a finite ${\mathcal{W}}_n\subseteq {\mathcal{L}}_n$. Then, as above,

$$\mathfrak{R}\subseteq \bigcup _n\bigcup _{L\in {\mathcal{W}}_n}L,$$

so $\left({\mathcal{W}}_n\right)$ is a sequence of finite selections covering $\mathfrak{R}$. Thus, $\mathfrak{R}$ satisfies ${\mathrm{S}}_{\mathrm{fin}}(\mathcal{D},\mathcal{D})$.

□

Definition 16. 

Point-D-open game $G_D\left(\mathfrak{E}\right)$: ONE plays D-covers ${\mathcal{L}}_n$; two responses $L_n\in {\mathcal{L}}_n$, two wins if $\left\{L_n\right\}$ D-covers $\mathfrak{E}$.

Example 9. 

Let $\mathfrak{E}=\mathbb{R}$ with the usual topology. Make a compact reduction modification.

$$K_n=[-n,n](n\in \mathbb{N}).$$

For each inning n, one plays the D-cover

$${\mathcal{L}}_n=\{\mathbb{R}\setminus K_n\}\cup \{G:G\mathit{open}\mathit{in}\mathbb{R}\mathit{and}{K}_n\subseteq G\}.$$

Each member of ${\mathcal{L}}_n$ is a D-set: opens are $L\setminus ⌀$, and $\mathbb{R}\setminus K_n$ is open since $K_n$ is compact (hence closed).

There are two replies at inning n by choosing any $L_n\in {\mathcal{L}}_n$ with $K_n\subseteq L_n$. Then,

$$\bigcup _{n\in \mathbb{N}}{L}_n\supseteq \bigcup _{n\in \mathbb{N}}{K}_n=\mathbb{R}.$$

Hence, $\{L_n:n\in \mathbb{N}\}$ D-covers $\mathfrak{E}$. Thus, there is a double victory in the present match run.

The above instance illustrates how a σ-compact two can convert because it is exhausted. D-covers of the form “outside $K_n$ or any open superset of $K_n$” into a covering sequence by selecting the compact cores’ open neighborhoods.

Proposition 2. 

If two has a winning strategy in $G_D\left(\mathfrak{E}\right)$ and $\Psi :\mathfrak{E}\to \mathfrak{R}$ is D-Lindelöf perfect onto, then two has a winning strategy in $G_D\left(\mathfrak{R}\right)$.

Proof. 

Assume two has a winning strategy $\sigma$ in $G_D\left(\mathfrak{E}\right)$. Define a strategy ${\rm Y}$ for two in $G_D\left(\mathfrak{R}\right)$ as follows.

In inning n, ONE plays a D-cover ${\mathcal{L}}_n$ of $\mathfrak{R}$. Create a reversal:

$${\Psi }^{-1}\left[{\mathcal{L}}_n\right]:=\{{\Psi }^{-1}\left(L\right):L\in {\mathcal{L}}_n\}.$$

Since $L=O_1\setminus O_2$ with $O_i$ open in $\mathfrak{R}$ implies ${\Psi }^{-1}\left(L\right)={\Psi }^{-1}\left(O_1\right)\setminus {\Psi }^{-1}\left(O_2\right)$ with ${\Psi }^{-1}\left(O_i\right)$ open in $\mathfrak{E}$, each ${\Psi }^{-1}\left(L\right)$ is a D-set in $\mathfrak{E}$. Because $\Psi$ is onto, $f^{-1}\left[{\mathcal{L}}_n\right]$ covers $\mathfrak{E}$.

Feed the history $\left({\Psi }^{-1}\left[{\mathcal{L}}_0\right],\dots ,{\Psi }^{-1}\left[{\mathcal{L}}_n\right]\right)$ to $\sigma$. Let $\sigma$ choose $T_n\in {\Psi }^{-1}\left[{\mathcal{L}}_n\right]$. Pick any $L_n\in {\mathcal{L}}_n$ with $T_n={\Psi }^{-1}\left(L_n\right)$, and set ${\rm Y}$’s move to be $L_n$.

We show that ${\rm Y}$ is winning. Let $\mathfrak{r}\in \mathfrak{R}$. Choose $\mathfrak{e}\in \mathfrak{E}$ with $\Psi \left(\mathfrak{e}\right)=\mathfrak{r}$. Since $\sigma$ is winning on $\mathfrak{E}$, there exists n with $\mathfrak{e}\in T_n={\Psi }^{-1}\left(L_n\right)$. Hence, $\mathfrak{r}=\Psi \left(\mathfrak{e}\right)\in L_n$. Thus, $\{L_n:n\in \mathbb{N}\}$ covers $\mathfrak{R}$, is a D-cover of $\mathfrak{R}$. Therefore, ${\rm Y}$ is a winning strategy for two in $G_D\left(\mathfrak{R}\right)$. □

Definition 17. 

The D-Lindelöf number $L_D\left(\mathfrak{E}\right)$ is the least κ such that every D-cover has a subcover of size $\le \kappa$. A family $\mathcal{N}$ is a D-network if for every D-open L and $\mathfrak{e}\in L$ there exists $N\in \mathcal{N}$ with $\mathfrak{e}\in N\subseteq L$. Let $n{w}_D\left(\mathfrak{E}\right)$ be the smallest size of a D-network.

Example 10. 

Let S be an uncountable set of size $\mathfrak{c}$ and fix $p\in S$. Give $\mathfrak{E}=S$ the Fortissimo topology: $L\subseteq \mathfrak{E}$ is open if either $L=⌀$ or $p\in L$ and $\mathfrak{E}\setminus L$ is countable. Then, the D-sets are exactly

$$\mathcal{D}\left(\mathfrak{E}\right)=\left\{\mathit{co}\text{−}\mathit{countable}\mathit{sets}\mathit{containing}p\right\}\cup \{\mathit{countable}\mathit{subsets}\mathit{of}\mathfrak{E}\setminus \{p\left\}\right\}\cup \{⌀\}.$$

Claim 3. 

$L_D\left(\mathfrak{E}\right)={\aleph }_0$.

Lower bound. Fix a countably infinite $C\subseteq \mathfrak{E}\setminus \left\{p\right\}$ and consider the D-cover 

$$\mathcal{L}=\{\mathfrak{E}\setminus C\}\cup \left\{\left\{\mathfrak{e}\right\}:\mathfrak{e}\in C\right\}.$$

Every subcover must include $\mathfrak{E}\setminus C$ to cover p and must include $\left\{\mathfrak{e}\right\}$ for each $\mathfrak{e}\in C$. Hence, any subcover has size ${\aleph }_0$. So $L_D\left(\mathfrak{E}\right)\ge {\aleph }_0$.

Upper bound. Let $\mathcal{T}$ be any D-cover of $\mathfrak{E}$. Some $T_0\in \mathcal{T}$ must contain p; hence, $T_0$ is co-countable. Its complement $C:=\mathfrak{E}\setminus T_0$ is countable. For each $\mathfrak{e}\in C$ choose $T_{\mathfrak{e}}\in \mathcal{T}$ with $\mathfrak{e}\in T_{\mathfrak{e}}$. Then 

$$\left\{T_0\right\}\cup \{T_{\mathfrak{e}}:\mathfrak{e}\in C\}$$

is a countable subcover. Thus, $L_D\left(\mathfrak{E}\right)\le {\aleph }_0$. After combining them, $L_D\left(\mathfrak{E}\right)={\aleph }_0$.

Claim 4. 

$n{w}_D\left(\mathfrak{E}\right)=\mathfrak{c}$.

Upper bound. Let $\mathcal{F}\subseteq {\left[\mathfrak{E}\right]}^{\le {\aleph }_0}$ be a cofinal family of countable subsets under ⊆ with $\left|\mathcal{F}\right|=\mathfrak{c}$. Set

$$\mathcal{N}:=\left\{\left\{\mathfrak{e}\right\}:\mathfrak{e}\in \mathfrak{E}\setminus \left\{p\right\}\right\}\cup \left\{\mathfrak{E}\setminus F:F\in \mathcal{F}\right\}.$$

We check the D-network property: If L is countable and $\mathfrak{e}\in L$, then $\left\{\mathfrak{e}\right\}\in \mathcal{N}$ and $\left\{\mathfrak{e}\right\}\subseteq L$.

If L is co-countable and $\mathfrak{e}\in L$: If $\mathfrak{e}\ne p$, again, $\left\{\mathfrak{e}\right\}\subseteq L$. If $\mathfrak{e}=p$, pick $F\in \mathcal{F}$ with $C\subseteq F$; then $N:=\mathfrak{E}\setminus F\in \mathcal{N}$, $p\in N$, and $N\subseteq \mathfrak{E}\setminus C=L$.

Hence, $\mathcal{N}$ is a D-network of size $\mathfrak{c}$, so $n{w}_D\left(\mathfrak{E}\right)\le \mathfrak{c}$.

Lower bound. Let $\mathcal{N}$ be any D-network. Consider

$${\mathcal{F}}_{\mathcal{N}}:=\left\{\mathfrak{E}\setminus N:N\in \mathcal{N},p\in N\right\}\subseteq {\left[\mathfrak{E}\right]}^{\le {\aleph }_0}.$$

For every countable $C\subseteq \mathfrak{E}$, we must have some $N\in \mathcal{N}$ with $p\in N\subseteq \mathfrak{E}\setminus C$; equivalently, $C\subseteq \mathfrak{E}\setminus N\in {\mathcal{F}}_{\mathcal{N}}$. Thus, ${\mathcal{F}}_{\mathcal{N}}$ is cofinal in ${\left[\mathfrak{E}\right]}^{\le {\aleph }_0}$. If $\left|\mathcal{N}\right|<\mathfrak{c}$, then $|{\mathcal{F}}_{\mathcal{N}}| <\mathfrak{c}$ and the union $L=\bigcup {\mathcal{F}}_{\mathcal{N}}$ has size $<\mathfrak{c}$. Choose a countably infinite $C\subseteq \mathfrak{E}\setminus L$. Then, no $D\in {\mathcal{F}}_{\mathcal{N}}$ contains C, contradicting cofinality. Hence, $\left|\mathcal{N}\right|\ge \mathfrak{c}$, so $n{w}_D\left(\mathfrak{E}\right)\ge \mathfrak{c}$.

Therefore $n{w}_D\left(\mathfrak{E}\right)=\mathfrak{c}$.

Proposition 3. 

For every $\mathfrak{E}$: $L\left(\mathfrak{E}\right)\le L_D\left(\mathfrak{E}\right)\le n{w}_D\left(\mathfrak{E}\right)$. If $\Psi :\mathfrak{E}\to \mathfrak{R}$ is D-Lindelöf perfect and onto, then $L_D\left(\mathfrak{R}\right)\le L_D\left(\mathfrak{E}\right)$.

Proof. 

(1) $L\left(\mathfrak{E}\right)\le L_D\left(\mathfrak{E}\right)$: Every open set L is a D-set ($L=L\setminus ⌀$), so every open cover is a D-cover. Any bound for D-covers bounds open covers.

(2) $L_D\left(\mathfrak{E}\right)\le n{w}_D\left(\mathfrak{E}\right)$: Let $\mathcal{N}$ be a D-network with $\left|\mathcal{N}\right|=n{w}_D\left(\mathfrak{E}\right)$. Given a D-cover $\mathcal{L}$ of $\mathfrak{E}$, define

$${\mathcal{N}}_{\mathcal{L}}:=\{N\in \mathcal{N}:\exists L\in \mathcal{L}\mathrm{with}N\subseteq L\}.$$

For each $N\in {\mathcal{N}}_{\mathcal{L}}$, choose and fix one $L_N\in \mathcal{L}$ with $N\subseteq L_N$. Then, $\{L_N:N\in {\mathcal{N}}_{\mathcal{L}}\}$ has size $\le \left|\mathcal{N}\right|$ and covers $\mathfrak{E}$: indeed, for any $\mathfrak{e}\in \mathfrak{E}$, pick $L\in \mathcal{L}$ with $\mathfrak{e}\in L$; by the D-network property, there is $N\in \mathcal{N}$ with $\mathfrak{e}\in N\subseteq L$, and hence, $\mathfrak{e}\in L_N$. Thus, every D-cover has a subcover of size $\le n{w}_D\left(\mathfrak{E}\right)$.

(3) If $\Psi :\mathfrak{E}\to \mathfrak{R}$ is onto and continuous, then $L_D\left(\mathfrak{R}\right)\le L_D\left(\mathfrak{E}\right)$: Let $\kappa =L_D\left(\mathfrak{E}\right)$. Given a D-cover $\mathcal{T}$ of $\mathfrak{R}$, the pullback

$${\Psi }^{-1}\left[\mathcal{T}\right]:=\{{\Psi }^{-1}\left(T\right):T\in \mathcal{T}\}$$

is a D-cover of $\mathfrak{E}$. Choose $\mathcal{W}\subseteq {\Psi }^{-1}\left[\mathcal{T}\right]$ with $\left|\mathcal{W}\right|\le \kappa$ covering $\mathfrak{E}$. For each $W\in \mathcal{W}$ pick $T_W\in \mathcal{T}$ with $W={\Psi }^{-1}\left(T_W\right)$. Then,

$$\mathfrak{R}=\Psi \left(\mathfrak{E}\right)\subseteq \Psi \left(\bigcup _{W\in \mathcal{W}}W\right)\subseteq \bigcup _{W\in \mathcal{W}}{T}_W,$$

so $\{T_W:W\in \mathcal{W}\}$ is a subcover of $\mathcal{T}$ of size $\le \kappa$. Hence, $L_D\left(\mathfrak{R}\right)\le L_D\left(\mathfrak{E}\right)$.

Combining (1) and (3) gives $L\left(\mathfrak{E}\right)\le L_D\left(\mathfrak{E}\right)\le n{w}_D\left(\mathfrak{E}\right)$ and the monotonicity under $\Psi$. □

Definition 18. 

Given a D-cover $\mathcal{L}$ and $A\subseteq \mathfrak{E}$, define ${\mathrm{St}}_{\mathcal{L}}\left(A\right)=\bigcup \{L\in \mathcal{L}:L\cap A\ne \varnothing \}$. $\mathfrak{E}$ is star-D-Lindelöf if there exists a countable $A\subseteq \mathfrak{E}$ such that ${\mathrm{St}}_{\mathcal{L}}\left(A\right)=\mathfrak{E}$ for every D-cover $\mathcal{L}$.

Example 11. 

Let $\mathfrak{E}=\mathbb{N}$ with the discrete topology. Then every D-set is open. Take the countable set $A=\mathfrak{E}$.

Claim: For every D-cover $\mathcal{L}$ of $\mathfrak{E}$, ${\mathrm{St}}_{\mathcal{L}}\left(A\right)=\mathfrak{E}$.

Indeed, since each $L\in \mathcal{L}$ satisfies $L\cap A=L\ne ⌀$, we have

$${\mathrm{St}}_{\mathcal{L}}\left(A\right)=\bigcup \{L\in \mathcal{L}:L\cap A\ne ⌀\}=\bigcup \mathcal{L}=\mathfrak{E}.$$

Thus, $\mathfrak{E}$ is star-D-Lindelöf with the fixed countable set $A=\mathfrak{E}$.

Sharpness in the discrete case: if $A⊊\mathfrak{E}$ is countable, the D-cover

$$\mathcal{L}=\{\mathfrak{E}\setminus A\}\cup \left\{\right\{a\}:a\in A\}$$

satisfies ${\mathrm{St}}_{\mathcal{L}}\left(A\right)=\bigcup \{\left\{a\right\}:a\in A\}=A\ne \mathfrak{E}$. Hence, no proper subset of $\mathfrak{E}$ can serve as a universal star core.

Theorem 13. 

If $\mathfrak{R}$ is paracompact and $\Psi :\mathfrak{E}\to \mathfrak{R}$ is D-Lindelöf perfect onto, then

$$\mathit{star}\text{−}D\text{−}\mathit{Lindel}\mathsf{ö}f\left(\mathfrak{E}\right)\Rightarrow \mathit{star}\text{−}D\text{−}\mathit{Lindel}\mathsf{ö}f\left(\mathfrak{R}\right).$$

Proof. 

Assume $\mathfrak{E}$ is star-D-Lindelöf. Fix a countable $A\subseteq \mathfrak{E}$ with ${\mathrm{St}}_{\mathcal{W}}\left(A\right)=\mathfrak{E}$ for every D-cover $\mathcal{W}$ of $\mathfrak{E}$. Let $B=\Psi \left[A\right]\subseteq \mathfrak{R}$; B is countable.

Let $\mathcal{L}$ be any D-cover of $\mathfrak{R}$. Its pullback

$${\Psi }^{-1}\left[\mathcal{L}\right]=\{{\Psi }^{-1}\left(L\right):L\in \mathcal{L}\}$$

is a D-cover of $\mathfrak{E}$. Hence,

$$\mathfrak{E}={\mathrm{St}}_{{\Psi }^{-1}\left[\mathcal{L}\right]}\left(A\right)=\bigcup \{{\Psi }^{-1}\left(L\right):L\in \mathcal{L},{\Psi }^{-1}\left(L\right)\cap A\ne ⌀\}.$$

Apply $\Psi$:

$$\mathfrak{R}=\Psi \left(\mathfrak{E}\right)\subseteq \bigcup \{L\in \mathcal{L}:{\Psi }^{-1}\left(L\right)\cap A\ne ⌀\}.$$

But ${\Psi }^{-1}\left(L\right)\cap A\ne ⌀$ iff $L\cap \Psi \left[A\right]\ne ⌀$, so

$$\mathfrak{R}\subseteq \bigcup \{L\in \mathcal{L}:L\cap B\ne ⌀\}={\mathrm{St}}_{\mathcal{L}}\left(B\right).$$

Since $\mathcal{L}$ was arbitrary, $\mathfrak{R}$ is star-D-Lindelöf. □

Theorem 14. 

Let $\mathfrak{E}$ be D-Lindelöf and $\mathfrak{R}$ be σ-compact. Then, $\mathfrak{E}\times \mathfrak{R}$ is D-Lindelöf. More generally, if $\mathfrak{R}={\bigcup }_n{K}_n$ with each $K_n$ compact, any D-cover of $\mathfrak{E}\times \mathfrak{R}$ admits a countable subcover.

Proof. 

Let K be compact and $\mathcal{L}$ a D-cover of $\mathfrak{E}\times K$, with $\mathfrak{E}$ D-Lindelöf. Fix $\mathfrak{e}\in \mathfrak{E}$. For each $\mathfrak{r}\in K$ choose $L_{\mathfrak{e},\mathfrak{r}}\in \mathcal{L}$ with $(\mathfrak{e},\mathfrak{r})\in L_{\mathfrak{e},\mathfrak{r}}$ and write $L_{\mathfrak{e},\mathfrak{r}}=O_{\mathfrak{e},\mathfrak{r}}\setminus T_{\mathfrak{e},\mathfrak{r}}$ with $O_{\mathfrak{e},\mathfrak{r}},T_{\mathfrak{e},\mathfrak{r}}$ open in $\mathfrak{E}\times K$. Since $T_{\mathfrak{e},\mathfrak{r}}$ is open and $(\mathfrak{e},\mathfrak{r})\notin T_{\mathfrak{e},\mathfrak{r}}$, there exist basic rectangles $W_{\mathfrak{e},\mathfrak{r}}\times N_{\mathfrak{e},\mathfrak{r}}\subseteq O_{\mathfrak{e},\mathfrak{r}}\setminus T_{\mathfrak{e},\mathfrak{r}}\subseteq L_{\mathfrak{e},\mathfrak{r}}$ containing $(\mathfrak{e},\mathfrak{r})$. Then, $\{N_{\mathfrak{e},\mathfrak{r}}:\mathfrak{r}\in K\}$ is an open cover of K, so by compactness, pick ${\mathfrak{r}}_1,\dots ,{\mathfrak{r}}_m$ with $K\subseteq {\bigcup }_{i=1}^m{N}_{\mathfrak{e},{\mathfrak{r}}_i}$ and set

$$W_{\mathfrak{e}}:=\bigcap _{i=1}^m{W}_{\mathfrak{e},{\mathfrak{r}}_i}(\mathrm{open}\mathrm{in}\mathfrak{E}),F_{\mathfrak{e}}:=\{L_{\mathfrak{e},{\mathfrak{r}}_i}:1\le i\le m\}.$$

We have

$$W_{\mathfrak{e}}\times K\subseteq \bigcup _{i=1}^m\left(W_{\mathfrak{e},{\mathfrak{r}}_i}\times N_{\mathfrak{e},{\mathfrak{r}}_i}\right)\subseteq \bigcup F_{\mathfrak{e}}\subseteq \bigcup \mathcal{L}.$$

Hence, $\{W_{\mathfrak{e}}:\mathfrak{e}\in \mathfrak{E}\}$ is an open (thus D-) cover of $\mathfrak{E}$. Since $\mathfrak{E}$ is D-Lindelöf, choose a countable subfamily ${\left\{W_{{\mathfrak{e}}_j}\right\}}_{j\in \mathbb{N}}$ covering $\mathfrak{E}$. Then, ${\bigcup }_j{F}_{{\mathfrak{e}}_j}$ is a countable subfamily of $\mathcal{L}$ covering $\left({\bigcup }_j{W}_{{\mathfrak{e}}_j}\right)\times K=\mathfrak{E}\times K$. Therefore, every D-cover of $\mathfrak{E}\times K$ admits a countable subcover.

Let $\mathfrak{R}={\bigcup }_n{K}_n$ with each $K_n$ being compact and let $\mathcal{L}$ be a D-cover of $\mathfrak{E}\times \mathfrak{R}$. For each n, $\mathcal{L}$ restricts to a D-cover of $\mathfrak{E}\times K_n$, so there is a countable subfamily ${\mathcal{L}}_n\subseteq \mathcal{L}$ covering $\mathfrak{E}\times K_n$. Then, ${\bigcup }_n{\mathcal{L}}_n$ is countable and covers $\mathfrak{E}\times \left({\bigcup }_n{K}_n\right)=\mathfrak{E}\times \mathfrak{R}$. □

Proposition 4. 

For a topological sum $\mathfrak{E}={⨁}_{i\in I}{\mathfrak{E}}_i$,

$$L_D\left(\mathfrak{E}\right)=max\left\{\left|I\right|,\underset{i\in I}{sup}{L}_D\left({\mathfrak{E}}_i\right)\right\}.$$

Proof. 

Lower bound.

(1)

The family $\{{\mathfrak{E}}_i:i\in I\}$ is a D-cover of $\mathfrak{E}$, so any subcover has size $\ge \left|I\right|$. Hence, $L_D\left(\mathfrak{E}\right)\ge \left|I\right|$.

(2)

Fix $i_0\in I$. Take a D-cover $\mathcal{L}$ of ${\mathfrak{E}}_{i_0}$ whose minimal subcover size is $L_D\left({\mathfrak{E}}_{i_0}\right)$. Then,

$$\mathcal{T}:=\{L\cup (\mathfrak{E}\setminus {\mathfrak{E}}_{i_0}):L\in \mathcal{L}\}$$

is a D-cover of $\mathfrak{E}$, and any subcover of $\mathcal{T}$ projects to a subcover of $\mathcal{L}$. Thus, $L_D\left(\mathfrak{E}\right)\ge L_D\left({\mathfrak{E}}_{i_0}\right)$. Taking the supremum over $i_0$ gives $L_D\left(\mathfrak{E}\right)\ge {sup}_i{L}_D\left({\mathfrak{E}}_i\right)$.

Upper bound. Let $\mathcal{W}$ be a D-cover of $\mathfrak{E}$. For each i, the restricted family

$$\mathcal{W}\upharpoonright {\mathfrak{E}}_i:=\{W\cap {\mathfrak{E}}_i:W\in \mathcal{W}\}$$

is a D-cover of ${\mathfrak{E}}_i$. Choose ${\mathcal{A}}_i\subseteq \mathcal{W}$ of size $\le L_D\left({\mathfrak{E}}_i\right)$ with $\{W\cap {\mathfrak{E}}_i:W\in {\mathcal{A}}_i\}$ covering ${\mathfrak{E}}_i$. Then ${\bigcup }_{i\in I}{\mathcal{A}}_i\subseteq \mathcal{W}$ covers $\mathfrak{E}$ and has cardinal

$$\le \sum _{i\in I}{L}_D\left({\mathfrak{E}}_i\right)\le \left|I\right|·\underset{i}{sup}{L}_D\left({\mathfrak{E}}_i\right)=max\left\{\right|I|,\underset{i}{sup}{L}_D\left({\mathfrak{E}}_i\right)\},$$

wherein the last equivalence is easy if all are finite and there is basic cardinal mathematics for infinite cardinals. Therefore, $L_D\left(\mathfrak{E}\right)\le max\left\{\right|I|,{sup}_i{L}_D\left({\mathfrak{E}}_i\right)\}$. □

7. Statistical Applications of $\mathit{D}$-Lindelöf Perfect Functions

Theorem 15. 

Let $f:X\stackrel{onto}{\to }Y$ be D-Lindelöf perfect, where $X,Y$ are Hausdorff spaces, with every $f^{-1}\left(y\right)$ being D-Lindelöf. Assume X is D-Lindelöf. Let $\mathcal{G}\subseteq \mathcal{B}\left(Y\right)$ be a collection such that $\{f^{-1}\left(G\right):G\in \mathcal{G}\}$ is a D-cover of X. Then there exists a ${\left\{G_n\right\}}_{n\in \mathbb{N}}\subseteq \mathcal{G}$ with ${\bigcup }_n{G}_n=Y$.

If, moreover, μ is a Radon probability on X, then $\nu :=f_{\#}\mu$ is tight on Y. Consequently, any countable $\left\{G_n\right\}$ is uniform Glivenko–Cantelli under ν.

Proof. 

Since X is D-Lindelöf and $\{f^{-1}\left(G\right):G\in \mathcal{G}\}$ is a D-cover of X, there exists $\left\{G_n\right\}$ with $X={\bigcup }_n{f}^{-1}\left(G_n\right)$. As f is onto,

$$Y=f\left(X\right)=f\left(\bigcup _n{f}^{-1}\left(G_n\right)\right)\subseteq \bigcup _n{G}_n\subseteq Y,$$

so ${\bigcup }_n{G}_n=Y$.

For tightness, let $\epsilon >0$. Since $\mu$ is Radon, choose a compact $K_X\subset X$ with $\mu \left(K_X\right)\ge 1-\epsilon$. A perfect f maps compact sets to compact sets, so $K_Y:=f\left(K_X\right)$ is compact in Y. Then

$$\nu \left(K_Y\right)=\mu \left(f^{-1}\left(K_Y\right)\right)\ge \mu \left(K_X\right)\ge 1-\epsilon ,$$

so $\nu$ is tight. Finally, any countable measurable $\left\{G_n\right\}$ is uniform Glivenko–Cantelli by the strong law on a countable collection and a diagonal argument. □

Proposition 5. 

Let X be D-Lindelöf and $L:X\times \Theta \to \mathbb{R}$ be lower semicontinuous in x for each $\theta \in \Theta$. For $a>b$, define D-sets

$$A(\theta ;a,b):=\{x:L(x,\theta )<a\}\setminus \{x:L(x,\theta )<b\}.$$

Suppose $\left\{A\right(\theta ;a,b):\theta \in \Theta \}$ covers X. Then there exists a countable ${\Theta }_0\subset \Theta$ with $\{A(\theta ;a,b):\theta \in {\Theta }_0\}$ covering X. In particular, the induced set class on X admits a countable bracket skeleton, yielding separability of the empirical process indexed by $\theta \mapsto {\mathbf{1}}_{\left\{L\right(·,\theta )<t\}}$ for each t.

Example 12. 

Let $X={\mathbb{R}}^3$ with the usual topology. Consider a positive class region

$$\mathcal{R}=\bigcup _{k\in K}\left(B(c_k,R_k)\setminus B(c_k,r_k)\right),0\le r_k<R_k,$$

a union of spherical shells (D-sets). If these shells cover the positive samples, D-Lindelöf on X gives a countable subfamily covering them. With the perfect function $f(x,y,z)=(x,y)$, the induced decision region on $Y={\mathbb{R}}^2$ is ${\bigcup }_{k}f\left(B(c_k,R_k)\right)\setminus f\left(B(c_k,r_k)\right)$. According to Theorem 15, any measurable family on Y whose pullbacks D-cover X admits a countable covering family on Y, and tight pushforwards preserve learnability of countable decision classes.

Example 13. 

Suppose X be D-Lindelöf and $\left\{L\right(·,\theta ):\theta \in \Theta \}$ is a loss family. For fixed t and a grid $\left\{(a_j,b_j)\right\}$ with $a_j>b_j$, the D-family ${\mathcal{A}}_t=\{A(\theta ;a_j,b_j):\theta \in \Theta ,j\in \mathbb{N}\}$ covers X. Proposition 5 generates a countable ${\Theta }_0$ sufficient to bracket $\{x:L(x,\theta )<t\}$, providing a countable parameter skeleton for screening and uniform LLN on the resulting countable class.

D-covers formalize selectable decision regions. D-Lindelöf supplies countable reduction. Perfect maps transfer coverage properties to observation spaces while maintaining tightness, enabling uniform laws on countable pushforward classes and practical model selection by countable cores.

The visualization pipeline appears in Figure 1, Figure 2 and Figure 3: Figure 1 shows the dataset, Figure 2 displays the selected D-cover, and Figure 3 depicts the pushforward under $f(x,y,z)=(x,y)$.

8. Conclusions

We examined the connections among the topological spaces that functions arise, including perfect spaces and D-Lindelöf perfect functions. Using the previously introduced notion of D-Lindelöf perfect functions, we identified conditions that align D-sets with locally indiscrete spaces. We established structural and preservation properties, and we specified when D-covers admit countable reductions. We identified their main characteristics in general and made clear the requirements for establishing comparable links between them. We also formulated applications to statistics under explicit assumptions. Numerous instances were included to illustrate these points, and the results delineate immediate directions for further study. Prospective research may be conducted to further investigate variations of these functions. Furthermore, defining and studying D-Lindelöf perfect properties within bitopological spaces, defining and studying fuzzy D-Lindelöf perfect functions, and finding a use for our new results of D-Lindelöf perfect functions in data analysis, optimization, and computational topology are our future objectives. Looking ahead, this work opens several promising avenues for research: unifying Lindelöf-type properties with D-set operations to generalize compactness [17]; investigating compatibility between topologies governed by D-Lindelöfness [22]; and extending decomposition theorems for D-covers in manifold theory [25].