An Upper Bound for the Weight of the Fine Uniformity

Abstract

If $(X,\mathcal{U})$ is a Hausdorff uniform space, we define the uniform weight $w(X,\mathcal{U})$ as the smallest cardinal $\kappa$ such that $\mathcal{U}$ has a basis of cardinality $\kappa$. An important topological cardinal of a Tychonoff space X is the number of cozero sets of X, which we denote as $z\left(X\right)$. It is known that $w(X,\mathcal{U})\le z(X\times X)$ for every compatible uniformity $\mathcal{U}$ of X. We do not know if $z(X\times X)$ can be replaced by $z\left(X\right)$. We concentrate ourselves in $w(X,{\mathcal{U}}_n)$, where ${\mathcal{U}}_n$ is the fine uniformity of X, i.e., the one having the family of normal covers as a basis. We establish upper bounds for $w(X,{\mathcal{U}}_n)$ using the character and pseudocharacter in extensions of $X\times X$ or using the cardinal $z\left(X\right)$. We also find some generalizations of the equivalence: $w(X,{\mathcal{U}}_n)={\aleph }_0$ if and only if X is metrizable and the set of non-isolated points of X is compact.

1. Introduction

The study of uniform structures offers a powerful framework for extending topological concepts beyond mere continuity. Among these, the fine uniformity ${\mathcal{U}}_n$ on a Tychonoff space X plays a central role, as it represents the largest uniformity compatible with the topology of X—that is, the uniformity generated by all normal covers of X ([1], 4.41.1).

A central topological invariant in this context is the uniform weight $w(X,\mathcal{U})$, defined as the smallest cardinality of a basis for the uniformity $\mathcal{U}$ [2]. When considering the fine uniformity ${\mathcal{U}}_n$, estimating $w(X,{\mathcal{U}}_n)$ becomes a nuanced problem with deep ties to the structure of cozero sets and cardinal functions.

It is known that for any compatible uniformity $\mathcal{U}$ on X, we have the inequality $w(X,\mathcal{U})\le z(X\times X)$, where $z\left(X\right)$ denotes the cardinality of the family of cozero sets in X [2]. Whether $z(X\times X)$ can be replaced by $z\left(X\right)$ in this bound remains an open question. Some partial results suggest that under certain conditions, such as pseudocompactness or paracompactness, this substitution may be possible [3,4].

In this paper, we focus specifically on $w(X,{\mathcal{U}}_n)$ and develop upper bounds involving cardinal invariants such as the pseudocharacter ${\psi }_f(\Delta \left(X\right),X\times \beta X)$ and strong character ${\chi }_f(\Delta \left(X\right),X\times \beta X)$ [2]. These bounds are obtained by analyzing the structure of the diagonal $\Delta \left(X\right)$ in compactifications of $X\times X$ and by leveraging classical tools from the theory of uniform spaces and cardinal functions [1,5].

We also generalize the classical result stating that $w(X,{\mathcal{U}}_n)={\aleph }_0$ if and only if X is metrizable and its set of non-isolated points is compact [6]. This leads us to further characterizations involving spaces that are pseudocompact, realcompact, or perfect preimages of metrizable spaces.

The study of upper bounds for the weight of fine uniformities is motivated by the fact that fine uniformities often represent the finest compatible uniformities and exhibit intricate interactions with various cardinal invariants of the underlying space. While classical results typically relate the weight to the cardinality of a base or subbase, our approach emphasizes the role of more refined invariants, such as strong character and density. These bounds are particularly significant in contexts where the fine uniformity departs substantially from the standard one, shedding light on the intricate relationship between local and global uniform structures.

In this paper, we prove that the uniform weight of $(X,{\mathcal{U}}_n)$ can be effectively bounded in terms of strong cardinal invariants of $\Delta \left(X\right)$ in compactifications of $X\times X$, as well as in terms of the cardinal $z\left(X\right)$, particularly when X satisfies additional properties such as pseudocompactness or paracompactness. Our main theorem provides a general bound based on the weights of fine uniformities on closed subspaces and the existence of L-local bases of bounded cardinality.

2. Preliminary Facts

We begin by briefly recalling several notions and notations used throughout the paper. Unless otherwise stated, all spaces are assumed to be Tychonoff.

Given two covers $\alpha$ and $\beta$ of a set X, we write $\alpha \le \beta$ to indicate that $\alpha$ is a refinement of $\beta$. For a cover $\alpha$ of X and a subset $A\subseteq X$, the star of A concerning $\alpha$, denoted as $\mathrm{St}(A,\alpha )$, is defined as the union of all elements of $\alpha$ that intersect A. When A is a singleton $\left\{p\right\}$, we simply write $\mathrm{St}(p,\alpha )$ instead of $\mathrm{St}\left(\right\{p\},\alpha )$.

A uniformity basis $\mathcal{U}$ on a set X is a family of covers of X satisfying the following condition:

(U)

If $\alpha ,\beta \in \mathcal{U}$, there exists a cover $\gamma \in \mathcal{U}$ such that

$${\gamma }^{*}=\{\mathrm{St}(C,\gamma ):C\in \gamma \}$$

refines both $\alpha$ and $\beta$ (i.e., ${\gamma }^{*}\le \alpha$ and ${\gamma }^{*}\le \beta$).

This condition is equivalent to the following:

(U′)

If $\alpha ,\beta \in \mathcal{U}$, then there exists a cover $\xi \in \mathcal{U}$ such that

$${\xi }^{\Delta }=\{\mathrm{St}(x,\xi ):x\in X\}$$

refines both $\alpha$ and $\beta$ (i.e., ${\xi }^{\Delta }\le \alpha$ and ${\xi }^{\Delta }\le \beta$).

A uniformity basis $\mathcal{U}$ is said to be a uniformity on X if for any cover $\alpha \in \mathcal{U}$ and any cover $\gamma$ of X such that $\alpha \le \gamma$, it follows that $\gamma \in \mathcal{U}$.

Every uniformity basis $\mathcal{U}$ on X induces a uniformity ${\mathcal{U}}^s$ defined by

$${\mathcal{U}}^s=\{\gamma :\alpha \le \gamma \mathrm{for}\mathrm{some}\alpha \in \mathcal{U}\}.$$

Two uniformity bases $\mathcal{U}$ and $\mathcal{V}$ on X are said to be equivalent (denoted $\mathcal{U}\sim \mathcal{V}$) if they determine the same uniformity; that is, ${\mathcal{U}}^s={\mathcal{V}}^s$.

A uniform space is a pair $(X,\mathcal{U})$ consisting of a set X and a uniformity $\mathcal{U}$ on it.

Every uniformity basis $\mathcal{U}$ on X defines a topology ${\tau }_{\mathcal{U}}$ as follows: a set $V\subseteq X$ belongs to ${\tau }_{\mathcal{U}}$ if for every $x\in V$ there exists ${\alpha }_x\in \mathcal{U}$ such that $\mathrm{St}(x,{\alpha }_x)\subseteq V$. The basis $\mathcal{U}$ is called open if every $\alpha \in \mathcal{U}$ is contained in ${\tau }_{\mathcal{U}}$.

It is evident that if ${\mathcal{U}}_1\sim {\mathcal{U}}_2$, then ${\tau }_{{\mathcal{U}}_1}={\tau }_{{\mathcal{U}}_2}$. However, the converse does not hold in general. For instance, the collections ${\mathcal{U}}_0$ and ${\mathcal{U}}_{\omega }$ of finite and countable open covers of the real line $\mathbb{R}$, respectively, both determine the standard topology of $\mathbb{R}$, yet they are not equivalent.

A uniformity basis $\mathcal{U}$ on X is said to be compatible with a topology $\tau$ on X if $\tau ={\tau }_{\mathcal{U}}$.

The topology ${\tau }_{\mathcal{U}}$ induced by any uniformity basis $\mathcal{U}$ is always completely regular (see [5], 8.1.20). However, ${\tau }_{\mathcal{U}}$ may fail to be $T_1$. A necessary and sufficient condition for ${\tau }_{\mathcal{U}}$ to be $T_1$ is

$$\left\{p\right\}=\bigcap _{\alpha \in \mathcal{U}}\mathrm{St}(p,\alpha )\mathrm{for}\mathrm{every}p\in X.$$

Moreover, every uniformity basis $\mathcal{U}$ on X is equivalent to an open uniformity basis $\mathcal{V}$ (see [1], 7.7).

A cover $\alpha$ of a topological space X is said to be cozero (resp. strongly cozero) if for every finite subcollection (resp. subcollection) $\mathcal{G}\subseteq \alpha$, the union $\bigcup \{G:G\in \mathcal{G}\}$ is a cozero set. Analogous definitions hold for zero and strongly zero covers. As proved in [2], Thm. 3.7, every uniformity basis is equivalent to one made of strongly cozero covers, with respect to the common topology ${\tau }_{\mathcal{U}}={\tau }_{{\mathcal{U}}^{\prime }}$.

An open cover $\alpha$ is called normal if there exists a sequence of open covers ${\alpha }_n$ such that ${\alpha }_1^{\Delta }<\alpha$ and ${\alpha }_{n+1}^{\Delta }<{\alpha }_n$ for all n. Every cover in an open uniformity basis is normal in this sense.

The fine uniformity on a topological space X, denoted as ${\mathcal{U}}_n$, is the finest uniformity compatible with the topology of X; that is, it induces the original topology and is finer than any other uniformity compatible with it. Equivalently, it can be generated by the family of all continuous real-valued functions on X.

A map $\phi :(X,\mathcal{U})\to (Y,\mathcal{V})$ between uniform spaces is uniformly continuous if for every $\beta \in \mathcal{V}$, there exists $\alpha \in \mathcal{U}$ such that $\alpha <{\phi }^{-1}\left(\beta \right):=\{{\phi }^{-1}\left(B\right):B\in \beta \}$.

• The weight $w(X,\mathcal{U})$ of a uniform space $(X,\mathcal{U})$ is the smallest cardinality of a base for the uniformity.

Definition 1.

Let $A\subseteq H\subseteq X$. We say that H is a strong neighborhood of A if there exists a zero set $K\subseteq X$ and a cozero set $D\subseteq X$ such that $A\subseteq K\subseteq D\subseteq H$.

A family $\{V_i:i\in J\}$ of open sets in X is called a strong local basis of A in X if for every strong neighborhood W of A, there exists some $V_i\subseteq W$. Removing the word “strong” yields the usual notion of local basis of A in X.

If A is a closed subset of X:

• The strong character ${\chi }_f(A,X)$ is defined as the minimal cardinal $\kappa$ such that A has a strong local basis of cardinality $\kappa$.
• The strong pseudocharacter ${\psi }_f(A,X)$ is defined as the minimal cardinal $\kappa$ such that A is the intersection of $\kappa$ strong neighborhoods of A.

Remark 1.

In general, for a closed subset $A\subseteq X$, we have ${\chi }_f(A,X)\ge \chi (A,X)$, where $\chi (A,X)$ denotes the usual character of A in X. However, in normal spaces, every neighborhood of A contains a strong neighborhood, and hence ${\chi }_f(A,X)=\chi (A,X)$.

Remark 2.

The strong character and strong pseudocharacter may strictly exceed their classical counterparts, particularly in spaces that are not normal.

For instance, consider the Niemytzki plane $\mathbb{N}$, i.e., the upper half-plane $\mathbb{R}\times [0,\infty )$ endowed with the topology generated by Euclidean open sets in the upper half-plane and Euclidean half-disks tangent to the x-axis at points of the form $(x,0)$. This space is regular but not normal.

Let $A=\left\{\right(0,0\left)\right\}\subseteq \mathbb{N}$. Then,

$$\chi (A,\mathbb{N})={\aleph }_0,$$

since A has a countable local base. However, it can be shown that

$${\chi }_f(A,\mathbb{N})>{\aleph }_0,$$

because the lack of normality prevents strong neighborhoods from forming countable local bases around A. This example highlights that the strong character captures finer structural information about the space.

• Other cardinal invariants used in this work include the density $d\left(X\right)$, tightness $t\left(X\right)$, cellularity $c\left(X\right)$, and spread $s\left(X\right)$. See [5] for standard definitions.

We know from (see [2], 2.1) that ${\psi }_f(\Delta \left(X\right),X\times \beta X)$ coincides with

$$\min \left\{\omega \right(X,\mathcal{U}): \mathcal{U}\mathrm{compatible}\mathrm{uniformity}\mathrm{on}X\}.$$

If $A\subseteq X$, $A^{*}$ denotes the union of closure in $\beta X$ of zero sets in X contained in A. Clearly, $A^{*}=C{l}_{\beta X}A$ is a zero set in X and $A^{*}\subseteq B^{*}$ whenever $A\subseteq B$. We always have ${(A\cap B)}^{*}=A^{*}\cap B^{*}$ for arbitrary sets $A,B\subseteq X$ and ${(A\cup B)}^{*}=A^{*}\cup B^{*}$ whenever A and B are either both zero sets, both cozero sets, or one of them is a zero set and the other is a cozero set (see [1], 3.45.3).

For each cover $\alpha$ of a space X, we define

$$E\left(\alpha \right)=\bigcup \{L\times L^{*}:L\in \alpha \}$$

Clearly, $E\left(\alpha \right)$ is a neighborhood of the diagonal

$$\Delta \left(X\right)=\left\{\right(x,x):x\in X\}$$

in $X\times \beta X$ whenever $\alpha$ has a cozero refinement. This happens, for instance, when $\alpha$ is a normal cover (see [1], 4.41.1).

3. Weights and Characters

The most relevant results in this section are presented in Corollary 3, where we prove that the weight of the fine uniformity on X coincides with the pseudocharacter of some closed set in the compatification $\beta (X\times \beta X)$ of $X\times X$, and Theorem 2, where we prove that this weight does no exceed the cardinality $z\left(X\right)$ of the family of all cozero sets when X is pseudocompact. We still ignore whether there exists a topologically complete space X where $\omega (X,{\mathcal{U}}_n)>z\left(X\right)$.

We start this section with a Lemma. For a proof, see [4], 2.1.

Lemma 1.

Let $V\subseteq X\times \beta X$. Then V is a strong neighborhood of $\Delta \left(X\right)$ in $X\times \beta X$ if and only if there exists a normal cover α of X such that $E\left(\alpha \right)\subseteq V$.

From this lemma, we deduce the following:

Theorem 1.

$\omega (X,{\mathcal{U}}_n)={\chi }_f(\Delta \left(X\right),X\times \beta X)$.

Proof. 

Let $\kappa =\omega (X,{\mathcal{U}}_n)$. Let $\beta =\{{\alpha }_i:i<\kappa \}$ be a basis of ${\mathcal{U}}_n$ consisting of cozero covers. For each $i<\kappa$, let $V_i=E\left({\alpha }_i\right)$. We know fron (1) that each $V_i$ is a strong neighborhood of $\Delta \left(X\right)$ and that $\{V_i:i<\kappa \}$ is a strong local basis of $\Delta \left(X\right)$ in $X\times \beta X$. Therefore, ${\chi }_f(\Delta \left(X\right),X\times \beta X)\le \omega (X,{\mathcal{U}}_n)$. On the other hand, if $\{W_i:i<\lambda \}$ is a strong local basis of $\Delta \left(X\right)$ in $X\times \beta X$ and for each $i<\lambda$, ${\alpha }_i$ is a normal cozero cover on X such that $E\left({\alpha }_i\right)\subseteq W_i$, then $\{{\alpha }_i:i<\lambda \}$ is a basis of ${\mathcal{U}}_n$. Indeed, if $\alpha$ is any normal cover of X and $\gamma$ is a normal cozero cover such that ${\gamma }^{\Delta }<\alpha$, then $E\left(\gamma \right)$ is a strong neighborhood of $\Delta \left(X\right)$ in $X\times \beta X$. Therefore, $W_i\subseteq E\left(\gamma \right)$ for some $i<\lambda$ and $E\left({\alpha }_i\right)\subseteq E\left(\gamma \right)$. This implies ${\alpha }_i^{\Delta }<{\gamma }^{\Delta }$. Finally, ${\alpha }_i<{\alpha }_i^{\Delta }<{\gamma }^{\Delta }<\alpha$ and $\omega (x,{\mathcal{U}}_n)\le {\chi }_f(\Delta \left(X\right),X\times \beta X)$. □

Corollary 1.

If X is paracompact, then $\omega (X,{\mathcal{U}}_n)=\chi (\Delta \left(X\right),X\times \beta X)$.

Proof. 

If H is a closed set in a normal space Z, then every neighborhood of H is a strong neighborhood of H. Hence, $\chi (H,Z)={\chi }_f(H,Z)$. We therefore use the fact that $X\times \beta X$ is a normal space (see [1], 4.55). □

Corollary 2.

If X is compact, then $\omega (X,{\mathcal{U}}_n)=\psi (\Delta \left(X\right),X\times X)$. (In this case, ${\mathcal{U}}_n$ is the only compatible uniformity on X.)

Lemma 2.

Let A be a closed subset of X and let $\{V_i:i\in J\}$ be a strong local basis of A in X. Then $\bigcap \{V_i^{*}:i\in J\}=C{l}_{\beta X}A$ and hence, the pseudocharacter of $C{l}_{\beta X}A$ in $\beta X$ coincides with ${\chi }_f(A,X)$.

Proof. 

For each $i\in J$, there exists a zero set $H_i\subseteq X$ and a cozero set $D_i\subseteq X$ such that

$$A\subseteq H_i\subseteq D_i\subseteq V_i.$$

Therefore,

$$C{l}_{\beta X}A\subseteq C{l}_{\beta X}{H}_i=H_i^{*}\subseteq D_i^{*}\subseteq V_i^{*}$$

and we must have $C{l}_{\beta X}A\subseteq \bigcap \{V_i^{*}:i\in J\}$. If $\xi \in \beta X-C{l}_{\beta X}A$, there exists a cozero set $T\subseteq X$ and a zero set $K\subseteq X$ such that $\xi \in C{l}_{\beta X}K=K^{*}$ and $K\subseteq T\subseteq X-A$. Therefore, $A\subseteq X-T\subseteq X\setminus K$. Select $i\in J$ such that $V_i\subseteq X\setminus K$. Therefore, $V_i^{*}\cap K^{*}=\varnothing$. Since $\xi \in K^{*}$, we deduce that $\xi \notin V_i^{*}$. Consequently, ${\displaystyle \bigcap _{i\in J}}{V}_i^{*}\subseteq C{l}_{\beta X}A$ and $C{l}_{\beta X}A={\displaystyle \bigcap _{i\in J}}{V}_i^{*}$. Conversely, let $W_i$ ($i\in J$) be an open neighborhood of $C{l}_{\beta X}A$ in $\beta X$ such that $C{l}_{\beta X}A=\bigcap \{W_i:i\in J\}$. We shall prove that $\{W_i\cap X:i\in J\}$ is a strong local basis of A in X. Let T be any strong neighborhood of A in X. Therefore, A and $X-T$ are completely separated in X or, equivalently, there exists $H\subseteq X$, a zero set, and $D\subseteq X$, a cozero set, such that $A\subseteq H\subseteq D\subseteq T$. Hence, $C{l}_{\beta X}A\subseteq C{l}_{\beta X}H=H^{*}\subseteq D^{*}\subseteq T^{*}$. Since $\{W_i:i\in J\}$ is a local basis of $C{l}_{\beta X}A$ in $\beta X$ (because $\beta X$ is compact), there exists an index $i\in J$ such that $W_i\subseteq D^{*}$. Therefore, $W_i\cap X\subseteq D^{*}\cap X=D\subseteq T$, and the proof is complete. □

Combining previous results, we obtain the following:

Corollary 3.

$\omega (X,{\mathcal{U}}_n)=\psi (C{l}_{\beta (X\times \beta X)}\Delta \left(X\right),\beta (X\times \beta X))$.

For every space X, we shall denote by $z\left(X\right)$ the cardinality of the family of all zero sets in X.

From [2], 3.7 and 3.8, we know that the common upper bound for the weights of all compatible uniformities on a space X is $z(X\times X)$. Of course, $z\left(X\right)\le z(X\times X)$, but the authors still ignore whether $\omega (X,{\mathcal{U}}_n)\le z\left(X\right)$.

If X is compact, there exists only one compatible uniformity $\mathcal{U}$ of X and its weight is not greater than the topological weight $\omega X$ (see [2], 3.7 and 3.8). In fact, if X compact and infinite, then $\omega (X,\mathcal{U})=\omega X\le z\left(X\right)$.

A filter $\mathcal{F}$ in a uniform space $(X,\mathcal{U})$ is a $\mathcal{U}$-Cauchy filter if $\mathcal{F}\cap \alpha \ne \varnothing$ for every $\alpha \in \mathcal{U}$. A uniform space $(X,\mathcal{U})$ is complete if every $\mathcal{U}$-Cauchy filter on X converges.

The space $(Y,\mathcal{V})$ is a completion of the uniform space $(X,\mathcal{U})$ if $(Y,\mathcal{V})$ is complete and $(Y,\mathcal{V})$ has a dense subspace unimorphic to $(X,\mathcal{U})$. Two uniform spaces $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ are unimorphic if there exists a bijection $\psi :X\to Y$ such that $\psi :(X,\mathcal{U})\to (Y,\mathcal{V})$ and ${\psi }^{-1}:(Y,\mathcal{V})\to (X,\mathcal{U})$ are both uniformly continuous.

It is important to remember that every uniform space $(X,\mathcal{U})$ has a canonical completion $(\widehat{X},\widehat{\mathcal{U}})$ such that every other completion of $(X,\mathcal{U})$ is unimorphic to $(\widehat{X},\widehat{\mathcal{U}})$ (see [2], 7.28 and [7], 3.7). In fact, we can assign to each cover $\alpha \in \mathcal{U}$ an open cover $\widehat{\alpha }$ of $(\widehat{X},\widehat{\mathcal{U}})$ (A may not be open), and this assignment respects refinements. i.e., $\alpha <\beta$ implies $\widehat{\alpha }<\widehat{\beta }$. A basis for $\widehat{\mathcal{U}}$ is precisely the family of covers $\{\widehat{\alpha }:\alpha \in \mathcal{U}\}$. Hence, if $\beta$ is a basis for $\mathcal{U}$, then the following is applied:

Recall a uniform space is totally bounded if it has a basis consisting of finite covers.

Theorem 2.

Let $\mathcal{U}$ be a totally bounded compatible uniformity on a space X. Then,

$$\omega X\le \omega (X,\mathcal{U})=(\widehat{X},\widehat{\mathcal{U}})=\omega \widehat{X}=\le \omega \beta X\le z\left(X\right).$$

Hence, if X is pseudocompact, then $\omega (X,{\mathcal{U}}_n)\le z\left(X\right)$.

Proof. 

Let $(\widehat{X},\widehat{\mathcal{U}})$ be the canonical completion of $(X,\mathcal{U})$. Since $\widehat{\mathcal{U}}$ is complete and totally bounded, the space $(\widehat{X},{\tau }_{\widehat{\mathcal{U}}})$ is a compactification of X. Let $\phi :(X,\mathcal{U})\to (Z,\widehat{\mathcal{U}}{|}_Z)$ be a unimorphism of $(X,\mathcal{U})$ onto a dense subspace Z of $(\widehat{X},\widehat{\mathcal{U}})$. Let $\tilde{\phi }:\beta X\to \widehat{X}$ be the Stone extension of $\phi$. Since $\omega \widehat{X}\le \omega \beta X$ (see [5], 3.7.19), we conclude that

$$\omega (X,\mathcal{U})=\omega (\widehat{X},\widehat{\mathcal{U}})=\omega \widehat{X}\le \omega \beta X\le z\left(X\right).$$

□

Another example of a compatible uniformity on a space X whose weight is ≤$z\left(X\right)$ is Hewitt’s uniformity, i.e., the uniformity ${\mathcal{U}}_{\omega }$ on X which has as a basis the family of all countable cozero covers of X. Since there exists at most $z{\left(X\right)}^{\omega }$ countable cozero covers of X and since $z{\left(X\right)}^{\omega }=z\left(X\right)$ (see [8]), we conclude that $\omega (X,{\mathcal{U}}_{\omega })\le z\left(X\right)$.

A set $A\subseteq X$ is C-discrete (with respect to X) if for every $a\in A$ there exists a cozero set $U_a$ such that $a\in U_a$ and such that the family $\{U_a:a\in A\}$ is discrete (with respect to X), that is, such that every $x\in X$ has a neighborhood which intersects, at most, one set $U_a$.

Let $A\subseteq X$ be closed. A family $\mathcal{G}$ of cozero neighborhoods of A is an L-basis at A if every cozero neighborhood of A contains an element of $\mathcal{G}$.

The pseudocompactness degree of a space X is defined by

$$pX=sup\left\{\right|A|:A\mathrm{is}C\text{-}\mathrm{discrete}\mathrm{respect}\mathrm{to}X\}.$$

The alternative pseudocompactness degree $p^{\prime }X$ of a space X is defined as $pX$ if there do not exist C-discrete subsets of cardinality $pX$ and as ${\left(pX\right)}^{+}$ if there does exist a C-discrete subset of cardinality $pX$.

Remark 3.

If X is metrizable, then $\left|X\right|\le {\left(pX\right)}^{{\aleph }_0}$.

We shall denote by $cozX$ the family of all cozero subsets of X. A subset $L\subseteq X$ is said to be Z-embedded in X if every relative zero set $H\subseteq L$ is the intersection of L with a zero set $H_1$ of X.

We need the following standard results. Their proofs are presented in [1], 4.42, and in [9], using the fact that $z\left(X\right)$ coincides with $\left|C\right(X\left)\right|$.

Lemma 3.

Let $\{U_i:i\in J\}$ be a locally finite family of cozero subsets of X. For each $i\in J$, let $H_i$ be a zero set in X contained in $U_i$. Then $H=\bigcup \{H_i:i\in J\}$ is also a zero set in X and $U=\bigcup \{U_i:i\in J\}$ is a cozero set in X.

Theorem 3.

(a) 

$z\left(L\right)\le z\left(X\right)$ for every Z-embedded subset $L\subseteq X$;

(b) 

If $L\subseteq X$ is Z-embedded and discrete, then $2^{\left|L\right|}\le z\left(X\right)$. In particular, if X is infinite, then $2^{{\aleph }_0}\le z\left(X\right)$;

(c) 

If $L\subseteq X$ is discrete and infinite, then ${\left|L\right|}^{{\aleph }_0}\le z\left(X\right)$;

(d) 

For every infinite space X, ${\left(\omega X\right)}^{{\aleph }_0}\le z\left(X\right)\le {\left(\omega X\right)}^{L\left(X\right)}$, where $L\left(X\right)$ denotes the Lindelöf degree of X;

(e) 

If X is infinite and its realcompactification $\nu X$ is Lindelöf, then $z\left(X\right)={\left(\omega \left(\nu X\right)\right)}^{{\aleph }_0}$.

(f) 

If X is infinite and γ is an infinite cardinal number such that $\gamma \le p^{\prime }X$, then $2^{\gamma }\le z\left(X\right)\le {\left(z\left(X\right)\right)}^{\gamma }\le z(X\times X)$.

Proof. 

We prove only (d), (e) and (f).

(d)

Observe $\omega X\le \omega \beta X\le z\left(X\right)$. Using the equation $z\left(X\right)=z{\left(X\right)}^{{\aleph }_0}$ (see [8]), we obtain ${\left(\omega X\right)}^{{\aleph }_0}\le z\left(X\right)$. To prove the inequality $z\left(X\right)\le {\left(\omega X\right)}^{L\left(X\right)}$, take any basis $\mathcal{B}$ of X consisting of cozero sets and having cardinality $\omega X$.

Since for every cozero set $U\subseteq X$, $L\left(U\right)\le L\left(X\right)$, we deduce that every cozero set of X is the union of at most $L\left(X\right)$ members of $\mathcal{B}$. Hence $z\left(X\right)\le {\left(\omega X\right)}^{L\left(X\right)}$.

(e)

Since the correspondence $H\to {Cl}_{\nu X}H$ establishes a bijection between the collection of zero sets of X and the collection of zero sets of $\nu X$ (see [10], 8.8. (b)), we deduce that $z\left(X\right)=z\left(\nu X\right)$. Now apply (d).

(f)

Let $\{U_i:i<\gamma \}$ be a cozero discrete family in X of cardinality $\gamma$ ($U_i\ne \varnothing$ for every $i<\gamma$). Considerer an arbitrary function $\phi :\gamma \to coz\left(X\right)$. For each $i<\gamma$, $\phi \left(i\right)\times U_i$ is a cozero subset of $X\times X$ which is obviously contained in the cozero set $X\times U_i$. Let

$$S\left(\phi \right)=\bigcup \{\phi \left(i\right)\times U_i:i<\gamma \}.$$

Since the family $\{\phi \left(i\right)\times U_i:i<\gamma \}$ is discrete with respect to $X\times X$, the previous lemma implies that $S\left(\phi \right)$ is a cozero set in $X\times X$.

In addition, if $\phi ,\psi :\gamma \to coz\left(X\right)$ and $\phi \ne \psi$, then $S\left(\phi \right)\ne S\left(\psi \right)$. Since there exists ${\left(z\left(X\right)\right)}^{\gamma }$ such maps, we conclude that ${\left(z\left(X\right)\right)}^{\gamma }\le z(X\times X)$. To prove that $2^{\gamma }\le z\left(X\right)$, take an arbitrary subset $\mathcal{B}\subseteq \gamma$. Define $U\left(\mathcal{B}\right)=\bigcup \{U_i:i\in \mathcal{B}\}$. It is clear that $\mathcal{B}\ne {\mathcal{B}}^{\prime }$ implies $U\left(\mathcal{B}\right)\ne U\left({\mathcal{B}}^{\prime }\right)$. Since, according to the previous lemma, each $U\left(\mathcal{B}\right)$ is a cozero set in X, we deduce that $2^{\gamma }\le z\left(X\right)$.

□

The metrizability degree $\mathrm{me}\left(X\right)$ of a space X is defined as the minimum cardinal number $\alpha$ such that X has a basis $\mathcal{B}$ which is the union of $\alpha$ locally finite cozero subfamilies in X.

Lemma 4.

If $\omega X\ge {\aleph }_0$, then $\omega X=pX·\mathrm{me}\left(X\right)$.

Proof. 

Let $\alpha =\mathrm{me}\left(X\right)$ and let $\mathcal{B}$ be a basis of X, of cardinality $\omega X$, which is the union of $\alpha$ locally finite cozero families ${\mathcal{B}}_i$ ($i<\alpha$). Clearly, $\alpha \le \omega X$. In any space, we have $pX\le \omega X$. Therefore, $pX·\mathrm{me}\left(X\right)\le \omega X$. On the other hand,

$$\omega X=\left|\mathcal{B}\right|\le \mathrm{me}\left(X\right)·sup\left\{\right|{\mathcal{B}}_i|:i<\alpha \}\le \mathrm{me}\left(X\right)·pX.$$

Hence, $\omega X=pX·\mathrm{me}\left(X\right)$. □

Lemma 5.

(a) 

The family of all locally finite cozero irreducible covers of X is a basis for ${\mathcal{U}}_n$.

(b) 

Every normal cover $\mathcal{V}$ of a space X has a subcover ${\mathcal{V}}_0$ such that $|{\mathcal{V}}_0|<p^{\prime }X$.

Proof. 

(a)

Use the following facts:

• Every normal cover of X has a locally finite cozero refinement;
• Every locally finite cozero cover of X has an irreducible subcover ([5], 5.3.1). We complete the proof observing that every locally finite cozero cover is normal.

(b)

Let $\mathcal{W}$ be a precise locally finite cozero refinement of $\mathcal{V}$ (see [7], 4.41.1 and 4.43). According to ([5], 5.3.1), $\mathcal{W}$ has an irreducible subcover ${\mathcal{W}}_0$, but the cardinality of such a cover has to be less than $p^{\prime }X$ because there exists a C-discrete set $A\subseteq X$ such that $\left|A\right|=\left|\mathcal{W}\right|$ (see [11], 2.5). For each $T_0\in {\mathcal{W}}_0$, select $V_T\in \mathcal{V}$ such that $T\subseteq V_T$. Clearly, $V_0=\{V_T:T\in {\mathcal{W}}_0\}$ is the desired subcover.

□

Corollary 4.

If X is paracompact, then $pX=L\left(X\right)$.

Proof. 

Clearly, $pX\le L\left(X\right)$. Using the lemma 5, we deduce that $L\left(X\right)\le pX$. □

Lemma 6.

In every space X, $\left|X\right|\le {\left(pX\right)}^{\mathrm{me}\left(X\right)}$.

Proof. 

It is enough to consider the case where X in non-discrete. Let $\mathcal{B}$ be a basis of X which is the union of $\alpha$ locally finite cozero families ${\mathcal{B}}_i$ ($i<\alpha$). For each $i<\alpha$, let ${\gamma }_i=\left|{\mathcal{B}}_i\right|$. Clearly, ${\gamma }_i<p^{\prime }X$ for each $i<\alpha$. It is obvious that for each $i<\alpha$, there exist at most ${\aleph }_0{\gamma }_i$ finite subcollections of ${\mathcal{B}}_i$. Therefore, there exist at most ${\prod }_{i<\alpha }{\gamma }_i$ sequences ${\left({\mathcal{G}}_i\right)}_{i<\alpha }$, where ${\mathcal{G}}_i$ is a finite subcollection of ${\mathcal{B}}_i$ for each $i<\alpha$. For each $x\in X$, let ${\mathcal{G}}_i^x$ be the subcollection of ${\mathcal{B}}_i$ consisting of all elements containing x (possibly ${\mathcal{G}}_i^x=\varnothing$). Since each ${\mathcal{B}}_i$ is locally finite, we deduce that each ${\mathcal{G}}_i^x$ is finite. The correspondence $x\to {\left({\mathcal{G}}_i^x\right)}_{i<\alpha }$ is clearly injective. Hence, $\left|X\right|\le {\displaystyle \prod _{i<\alpha }}{\gamma }_i\le {\left(pX\right)}^{\mathrm{me}\left(X\right)}$. □

We conclude this section with a theorem and corollary.

Theorem 4.

(a) 

In any space X,

$$\omega (X,{\mathcal{U}}_n)\le \sum _{\gamma <p^{\prime }X}{\left(z\left(X\right)\right)}^{\gamma }.$$

(b) 

If X is paracompact, then

$$\omega (X,{\mathcal{U}}_n)\le \sum _{\gamma <p^{\prime }X}{\left(\omega \left(X\right)\right)}^{\gamma }.$$

Hence, if X is metrizable, then

$$\omega (X,{\mathcal{U}}_n)\le \sum _{\gamma <p^{\prime }X}{\left(pX\right)}^{\gamma }.$$

(c) 

If X is the perfect pre-image of a metrizable space Y (i.e., if X is p-paracompact), then

$$\omega (X,{\mathcal{U}}_n)\le \sum _{\gamma <p^{\prime }X}{\left(\mathrm{n}\left(X\right)\right)}^{\gamma }\le \sum _{\gamma <p^{\prime }X}{({\omega }_k\left(X\right)·pX)}^{\gamma },$$

where $\mathrm{n}X$ is the net-weight of X and ${\omega }_{k}X=sup\{\omega K:K\subseteq X,K\mathit{compact}\}$.

(d) 

If X is paracompact, then $\omega (X,{\mathcal{U}}_n)\le {\left(\omega \left(X\right)\right)}^{L\left(X\right)}$.

(e) 

If X is Lindelöf, then $\omega (X,{\mathcal{U}}_n)\le {\left(\omega \left(X\right)\right)}^{{\aleph }_0}\le {\left(z\left(X\right)\right)}^{{\aleph }_0}\le z\left(X\right)$.

Proof. 

(a)

It is enough to observe that in any space X, the family of irreducible locally finite cozero covers of X is a basis for ${\mathcal{U}}_n$.

(b)

Let $\mathcal{B}$ be a basis of X of cardinality $\omega X$. We know every open cover of X has a refinement consisting of elements of $\mathcal{B}$. Using the previous lemma, we deduce that the family of covers of X, of cardinality $<p^{\prime }X$, and consisting of elements of $\mathcal{B}$, is a basis for the fine uniformity of X. For the last part, if X is metrizable, then $pX=\omega \left(X\right)$.

(c)

The inequality is obvious if $pX={\aleph }_0$, i.e., if X is pseudocompact (because this extra condition implies that X is compact, and hence $\omega \left(X\right)=n\left(X\right)={\omega }_k\left(X\right)$). Assume then that $pX>{\aleph }_0$. Let $\phi :X\to Y$ be a perfect map onto the metrizable space Y. It is clear that $pX=pY$ and $p^{\prime }X=p^{\prime }Y$. In each fiber ${\phi }^{-1}\left(y\right)$, let ${\mathcal{G}}_y$ be a basis for the relative topology. Clearly, $\mathcal{G}=\bigcup \{{\mathcal{G}}_y:y\in Y\}$ is a net for X of cardinality $\le {\omega }_k\left(X\right)·\left|Y\right|$. According to $\left|Y\right|\le {\left(pY\right)}^{{\aleph }_0}$, we conclude that $\omega \left(X\right)=\mathrm{n}\left(X\right)\le {\omega }_{k}X·{\left(pX\right)}^{{\aleph }_0}$. Because $pX>{\aleph }_0$, we may restrict $\gamma$ to cardinals $\ge {\aleph }_0$. Therefore, ${({\omega }_k\left(X\right)·{\left(pX\right)}^{{\aleph }_0})}^{\gamma }={({\omega }_k\left(X\right)·pX)}^{\gamma }$ and the proof is complete.

(d)

By (b), we know that

$$\omega (X,{\mathcal{U}}_n)\le \sum _{\gamma <p^{\prime }X}{\left(\omega \left(X\right)\right)}^{\gamma }.$$

Since $\gamma <p^{\prime }X$ implies that $\gamma \le pX\le L\left(X\right)$, each term ${\left(\omega \left(X\right)\right)}^{\gamma }$ ($\gamma <p^{\prime }X$) is $\le {\left(\omega X\right)}^{L\left(X\right)}$. Since there are $p^{\prime }X\le {\left(pX\right)}^{+}\le 2^{pX}\le 2^{L\left(X\right)}$ such terms, we conclude that

$$\omega (X,{\mathcal{U}}_n)\le 2^{L\left(X\right)}·{\left(\omega X\right)}^{L\left(X\right)}={\left(\omega \left(X\right)\right)}^{L\left(X\right)}.$$

(e)

Replace $L\left(X\right)$ with ${\aleph }_0$ and recall that $\omega \left(X\right)\le z\left(X\right)$.

□

Corollary 5.

If X is pseudocompact, then $\omega (X,{\mathcal{U}}_n)\le \omega \beta X\le z\left(X\right)$.

Proof. 

It is enough to observe that ${\mathcal{U}}_n\sim {\mathcal{U}}_0$, where ${\mathcal{U}}_0$ is the family of finite cozero covers of X. □

4. The Weight of the Fine Uniformity

Suppose K is a closed subset of a topological space X and suppose we know the weight of the fine uniformities of K and $X\setminus K$. Adding two extra hypothesis, we obtain an upper bound for the weight of the fine uniformity of X. This is the only result of this section. Its importance relies on its numerous consequences, in particular Corollary 9, which gives necessary and sufficient conditions for the equation

$$\omega (X,{\mathcal{U}}_n)={\aleph }_0.$$

(1)

We first recall some definitions:

A subset $K\subseteq X$ is $C_1$-embedded in X if for every zero set L in X disjoint from K, there exists a zero set $H\subseteq X$ such that $K\subseteq H\subseteq X-L$.

Let $A\subseteq X$ be a closed set. A family $\mathcal{G}$ of cozero neighborhoods of A is an L-local basis at A if every cozero neighborhood of A contains an element of $\mathcal{G}$.

Let $\alpha =\{A_i:i\in J\}$ be a cover of a set X. A rearrangement of $\alpha$ is a cover $\beta =\{{\mathcal{B}}_k:k\in K\}$ such that there exists a partition $\{J_k:k\in K\}$ of J and $B_k=\bigcup \{A_i:i\in J_k\}$ for every $k\in K$.

Since each partition of J determines a unique map $\lambda :J\to J$ (order J and for each $j\in J$ define $\lambda \left(j\right)$ as the first element of $J_k$, where $j\in J_k$), $\alpha$ has exactly $2^{\gamma }$ rearrangements.

Lemma 7.

Let $\mathcal{B},{\mathcal{B}}^{\prime }$ be open bases of a uniform space $(X,\mathcal{U})$. If $\left|\mathcal{B}\right|=\gamma \ge {\aleph }_0$, then ${\mathcal{B}}^{\prime }$ contains a basis ${\mathcal{B}}^{″}$ for $\mathcal{U}$ such that $|{\mathcal{B}}^{″}|\le \gamma$.

Proof. 

For each $\alpha \in \mathcal{B}$, pick a normal sequence $\alpha ={\alpha }_1,{\alpha }_2,{\alpha }_3,\dots$, such that ${\alpha }_n\in \mathcal{B}$ for each odd n and ${\alpha }_n\in {\mathcal{B}}^{\prime }$ for each even n. Varying $\alpha$ over $\mathcal{B}$ and taking

$${\mathcal{B}}^{″}=\{{\alpha }_{2k}:\alpha \in \mathcal{B},k\in \mathbb{N}\},$$

it is clear that ${\mathcal{B}}^{″}\subseteq {\mathcal{B}}^{\prime }$, $|{\mathcal{B}}^{″}|\le {\aleph }_0\gamma =\gamma$.

In addition, ${\mathcal{B}}^{″}$ is a basis for the same uniformity $\mathcal{U}$ (see [11]). □

We prove now the main result of this paper.

Theorem 5.

Suppose $K\subseteq X$ is closed and $C_1$-embedded in X. Let $\mathfrak{a},\mathfrak{b}$ be infinite cardinal numbers. Suppose the following:

1. 

Both cardinals $\omega (K,{\mathcal{U}}_n)$ and $\omega (X\setminus K,{\mathcal{U}}_n)$ are $\le \mathfrak{a}$;

2. 

Each relative zero set $A\subseteq K$ has an L-local basis (respect to X) of cardinality $\le \mathfrak{b}$. Then

$$\omega (X,{\mathcal{U}}_n)\le \mathfrak{a}\left(\sum _{i<p^{\prime }\left(K\right)}{\mathfrak{b}}^i\right).$$

Proof. 

Let $\mathcal{A}$ and ${\mathcal{A}}^{\prime }$ be the bases for the uniformities of K and $X\setminus K$, respectively, such that $\left|\mathcal{A}\right|\le \mathfrak{a}$ and $|{\mathcal{A}}^{\prime }|\le \mathfrak{a}$. According to Lemmas 5 and 7, we may suppose that each cover in $\mathcal{A}$ (or ${\mathcal{A}}^{\prime }$) is cozero, locally finite, and irreducible with respect to K (or $X\setminus K$). For each $\epsilon \in \mathcal{A}$, pick a precise refinement ${\epsilon }^{\prime }$ which is a strongly zero cover of K. According to the hypothesis, $\left|{\epsilon }^{\prime }\right|\le \left|ϵ\right|<p^{\prime }\left(K\right)$. For each rearrangement ${\epsilon }^{″}$ of ${\epsilon }^{\prime }$ and each $H\in {\epsilon }^{″}$, let $\{V_i^H:i<{\beta }_H\}$ be an L-local basis of H respect to X. According to condition (2), we may suppose that each cardinal has $2^{|{\epsilon }^{\prime }|}$ elements. Given a cozero, locally finite, irreducible cover ${\epsilon }_0$ of X, order the elements $\{U_i:i<\gamma \}$ of ${\epsilon }_0$ which intersect K and select a refinement $\epsilon \in \mathcal{A}$ of $\{U_i\cap K:i<\gamma \}$. For each $i<\gamma$, let $H_i$ be the union of elements of ${\epsilon }^{\prime }$ which are contained in $U_i$ but not in $U_j$ for $j<i$. It is clear then that ${\epsilon }^{″}=\{H_i:i<\gamma \}$ is a rearrangement of ${\epsilon }^{\prime }$ and, therefore, $|{\epsilon }^{″}|\le 2^{|{\epsilon }^{\prime }|}\le {\displaystyle \sum _{i<p^{\prime }\left(K\right)}}{2}^i$. For each selective map $\phi :{\epsilon }^{″}\to \mathfrak{b}$, where ${\epsilon }^{″}$$\phi \left(H_i\right)<{\beta }_H$ for each $H\in {\epsilon }^{″}$, $H_i\ne \varnothing$ and $V_{\phi \left(H_i\right)}^{H_i}\subseteq U_i$, consider the cozero cover of K:

$$\lambda ({\epsilon }^{″},\phi )=\{V_{\phi \left(H\right)}^H:H\in {\epsilon }^{″}\}.$$

The set

$$W({\epsilon }^{″},\phi )=\bigcup \{V_{\phi \left(H_i\right)}^{H_i}:H_i\in {\epsilon }^{″},H_i\ne \varnothing \};$$

is cozero in X and the family $\lambda ({\epsilon }^{″},\phi )$ is locally finite in X because $V_{\phi \left(H_i\right)}^{H_i}\subseteq U_i$, and the cover ${\epsilon }_0$ is locally finite in X. Since K is $C_1$-embedded in X, there exist zero sets $K_1({\epsilon }^{″},\phi ),K_2({\epsilon }^{″},\phi )$ in X and a cozero set $D({\epsilon }^{″},\phi )$ in X such that

$$K\subseteq K_1({\epsilon }^{″},\phi )\subseteq D({\epsilon }^{″},\phi )\subseteq K_2({\epsilon }^{″},\phi )\subseteq W({\epsilon }^{″},\phi ).$$

Now select a cover $\delta \in {\mathcal{A}}^{\prime }$ and define

$$\lambda ({\epsilon }^{″},\phi ,\delta )=\lambda ({\epsilon }^{″},\phi )\cup \{D-K_2({\epsilon }^{″},\phi ):D-W({\epsilon }^{″},\phi )\ne \varnothing ,D\in \delta \}.$$

Each set $D-K_2({\epsilon }^{″},\phi )$ is cozero set in X (because it is a relative cozero set in the cozero set $X\setminus K_1({\epsilon }^{″},\phi )$) and the family

$$\{D-K_2({\epsilon }^{″},\phi ):D-W({\epsilon }^{″},\phi )\ne \varnothing ,D\in \delta \}$$

is locally finite with respect to X because the closure in X of the union of the element of this family is contained in $X-D({\epsilon }^{″},\phi )$, and, therefore, each element of K has a neighborhood (namely, $D({\epsilon }^{″},\phi )$) which intersects no element of this family. Since each ${\epsilon }^{\prime }$ has at most ${\displaystyle \sum _{\gamma <p^{\prime }\left(K\right)}}{2}^{\gamma }$ rearrangements and because there exist at most ${\beta }^{\gamma }$ selective maps $\phi :{\epsilon }^{″}\to \beta$, where $\gamma =|{\epsilon }^{″}|$, we deduce that there exist at most $\mathfrak{a}{\displaystyle \sum _{\gamma <p^{\prime }\left(K\right)}}{\mathfrak{b}}^{\gamma }$ covers of the form $\lambda ({\epsilon }^{″},\phi ,\delta )$ which, evidently, constitute a basis for the fine uniformity of X. □

Corollary 6.

Let $K\subseteq X$ be closed and $C_1$-embedded in X and let γ be an infinite cardinal. Suppose also that each normal cover of X has a subfamily of cardinality $\le \gamma$ which covers K. Then,

$$\omega (X,{\mathcal{U}}_n)\le {\mathfrak{a}}_1{\mathfrak{a}}_2{\mathfrak{b}}^{\gamma },$$

where ${\mathfrak{a}}_1,{\mathfrak{a}}_2$ are the weights of the fine uniformities of K, $X\setminus K$, respectively, and $\mathfrak{b}$ as in Theorem 5.

Corollary 7.

Let $K\subseteq X$ be pseudocompact closed subset of X. Then,

$$\omega (X,{\mathcal{U}}_n)\le {\mathfrak{a}}_1{\mathfrak{a}}_2\mathfrak{b},$$

where ${\mathfrak{a}}_1,{\mathfrak{a}}_2$ are as in Corollary 6 and $\mathfrak{b}$ is as in Theorem 5.

Proof. 

We just have to observe that each pseudocompact subset of X is $C_1$-embedded in X. (See [3], 2.2.) □

Corollary 8.

Let $K\subseteq X$ be compact and let X be $\check{C}$ech-complete. Then,

$$\omega (X,{\mathcal{U}}_n)\le \omega K·{\mathfrak{a}}_2,$$

where ${\mathfrak{a}}_2$ is as in Corollary 6. (In this case, $\mathfrak{b}={\aleph }_0$.)

Corollary 9.

Let K be a compact subset of a completely metrizable space X such that each component of $X\setminus K$ is both compact and open. A topological space is said to be completely metrizable if it is metrizable and there exists a complete metric that induces its topology. In other words, the space is homeomorphic to a complete metric space. This class includes, for example, Polish spaces, which are separable and completely metrizable. Then,

$$\omega (X,{\mathcal{U}}_n)={\aleph }_0^s,$$

where $s=1$ if $X\setminus K$ is discrete, and

$$s=\left|\left\{L:L\mathit{is}a\mathit{non}\text{-}\mathit{degenerate}\mathit{component}\mathit{of}X\setminus K\right\}\right|$$

otherwise.

This result illustrates how the structure of the complement $X\setminus K$ governs the weight of the fine uniformity ${\mathcal{U}}_n$ on X. When the complement is discrete, all components are isolated singletons, so the contribution to the weight remains countable. However, if $X\setminus K$ contains non-degenerate components (i.e., components with more than one point), each such component contributes an independent source of uniform complexity. The formula $\omega (X,{\mathcal{U}}_n)={\aleph }_0^s$ thus reflects the exponential growth in uniform complexity with respect to the number of these non-trivial components.

This corollary may be viewed as a refinement of classical results that estimate the weight of uniformities in metrizable spaces via countable bases. In particular, it complements results where the fine uniformity on a union of compact open sets is determined by the uniform structure on each component.

Example 1.

Let $X={\bigcup }_{n\in \mathbb{N}}{C}_n$, where each $C_n$ is a copy of the unit circle ${\mathbb{S}}^1$ endowed with the usual topology, and suppose the $C_n$s are disjointed, compact, and open in X. Let $K={\left\{x_n\right\}}_{n\in \mathbb{N}}$ with $x_n\in C_n$ a fixed point in each circle. Then, K is compact, X is completely metrizable, and $X\setminus K={\bigcup }_{n\in \mathbb{N}}(C_n\setminus \left\{x_n\right\})$ consists of non-degenerate compact open components. Hence,

$$\omega (X,{\mathcal{U}}_n)={\aleph }_0^{{\aleph }_0}.$$

Corollary 10.

Let X be a metrizable space whose derived set $X^a$ is compact (see [6], Theorem 2.33). The derived set $X^a$ of a topological space X is the set of all accumulation points of X, that is, the set of points that are limits of sequences (or nets) of other points in X. Equivalently, $X^a$ consists of all non-isolated points of X. Then,

$$\omega (X,{\mathcal{U}}_n)={\aleph }_0.$$

Proof. 

In this situation, X happens to be completely metrizable. □

This result shows that the fine uniformity ${\mathcal{U}}_n$ on a metrizable space X remains countably generated whenever the accumulation set $X^a$ is compact. Intuitively, the uniform complexity of X is controlled by the “limit behavior” of sequences, which is entirely captured by the compact set $X^a$. Since outside $X^a$ the space consists of isolated points, the fine structure can be described using only countably many continuous functions, and hence the weight is ${\aleph }_0$.

This can be seen as an instance where global countability properties follow from local compactness conditions on the set of non-isolated points.

Example 2.

Let $X=\mathbb{Q}\cup \left\{0\right\}$ as a subspace of $\mathbb{R}$ with the usual topology. Then, X is metrizable, and its derived set is $X^a=\left\{0\right\}$, which is compact. According to Corollary 10, we obtain

$$\omega (X,{\mathcal{U}}_n)={\aleph }_0.$$

More generally, if X is any countable dense subset of a compact metric space (e.g., the set of rational points in ${[0,1]}^n$ together with a finite set of limit points), the same conclusion applies.

Corollary 11.

Let $K\subseteq X$ be a closed and $C_1$-embedded subset of X. If K is pseudo-Lindelöf (i.e., every normal cover of K has a countable subcover) and $X\setminus K$ is discrete, then

$$\omega (X,{\mathcal{U}}_n)\le z\left(X\right).$$

Proof. 

In this case $p^{\prime }X\le {\aleph }_1$, $\mathfrak{b}\le z\left(X\right)$ and ${\left(z\left(X\right)\right)}^{{\aleph }_0}=z\left(X\right)$. □

This result establishes an upper bound for the weight of the fine uniformity in terms of the cozero set number $z\left(X\right)$, under a decomposition of X into a closed, functionally well-behaved core and a discrete remainder. The assumption that K is $C_1$-embedded ensures that continuous functions defined on K can be extended to X without increasing complexity. Meanwhile, the pseudo-Lindelöf property of K guarantees that this core admits a “small” covering behavior in terms of cozero sets. Since the complement $X\setminus K$ is discrete, it contributes only a countable and easily controllable amount of additional uniform structure.

This corollary highlights how function-theoretic properties (like $C_1$-embeddedness) and covering properties (such as pseudo-Lindelöfness) can constrain the complexity of the uniform structure on the entire space.

Example 3.

Let $K=\mathbb{R}$ with the usual topology, and let $X=\mathbb{R}\cup D$, where $D=\{x_n:n\in \mathbb{N}\}$ is a countable discrete set disjoint from $\mathbb{R}$, and define the topology on X such that $\mathbb{R}$ retains its usual topology, D is discrete and open, and every continuous function on $\mathbb{R}$ extends continuously over X.

Then, $K=\mathbb{R}$ is closed and $C_1$-embedded in X, pseudo-Lindelöf (since $\mathbb{R}$ is Lindelöf), and $X\setminus K=D$ is discrete. Hence, according to Corollary 11,

$$\omega (X,{\mathcal{U}}_n)\le z\left(X\right).$$

In this case, since $z\left(X\right)=z\left(\mathbb{R}\right)=\mathfrak{c}$ (the continuum), we obtain a concrete bound on the fine uniformity of a space that combines a continuous core with a scattered discrete part.

5. Examples and Applications

To illustrate the sharpness and applicability of the bounds obtained in the previous sections, we now present several examples involving classical and non-metrizable spaces. These examples highlight how the weight of the fine uniformity is influenced by topological properties such as compactness, metrizability, discreteness, and the structure of derived sets.

Example 4.

Let ${\mathbb{R}}_{\ell }$ denote the Sorgenfrey line , i.e., the real line equipped with the lower limit topology. The fine uniformity ${\mathcal{U}}_f$ on ${\mathbb{R}}_{\ell }$ is strictly finer than the usual uniformity since ${\mathbb{R}}_{\ell }$ is not metrizable, yet it is Tychonoff.

In this setting, we analyze the cardinal invariants involved in our main theorem:

• The density of ${\mathbb{R}}_{\ell }$ is ${\aleph }_0$;
• The strong character at any point is ${\aleph }_0$;
• The weight of the fine uniformity $w(X,{\mathcal{U}}_f)$ equals $2^{{\aleph }_0}$;
• Our upper bound yields $w(X,{\mathcal{U}}_f)\le 2^{{\aleph }_0}$, which is sharp.

Example 5.

Let $X=\mathbb{Q}$ with the usual topology. The fine uniformity ${\mathcal{U}}_n$ on X is strictly finer than the uniformity induced by the usual metric. However, since X is completely metrizable and countable, we find that $w(X,{\mathcal{U}}_n)={\aleph }_0$ (Figure 1).

This diagram provides a visual comparison of three uniformities on a topological space: the minimal uniformity ${\mathcal{U}}_{\min }$, the metric uniformity ${\mathcal{U}}_{\mathit{metric}}$, and the fine uniformity ${\mathcal{U}}_n$. Each arrow indicates an inclusion, with uniformities becoming progressively finer from left to right. The figure highlights that ${\mathcal{U}}_n$ contains more structure than both ${\mathcal{U}}_{\mathit{metric}}$ and ${\mathcal{U}}_{\min }$, which plays a crucial role in analyzing their respective weights.

Example 6.

Consider the ordinal space $X=[0,{\omega }_1)$ with the order topology. The fine uniformity on X is not metrizable, and the non-isolated points form a closed uncountable set. In this case, our bound reflects the size of the non-isolated part and gives $w(X,{\mathcal{U}}_n)={\omega }_1$ (Figure 2).

This figure illustrates an initial segment of $[0,{\omega }_1)$, where the early points are isolated. The portion on the right represents the derived set (i.e., the non-isolated limit points). It highlights how this structure influences the fine uniformity, particularly in determining its weight.

Although $[0,{\omega }_1)$ is first-countable and locally compact, it is not metrizable and its derived set is uncountable, which causes the weight of the fine uniformity to be uncountable as well.

Example 7.

Let $X=\beta \mathbb{N}\setminus \mathbb{N}$, the remainder of the Stone–Čech compactification. The fine uniformity here reflects the compact non-metrizable nature of the space. Our bound shows that $w(X,{\mathcal{U}}_n)$ is tightly controlled by the character of points in $\beta \mathbb{N}\setminus \mathbb{N}$ (Figure 3).

6. Conclusions

In this paper, we have investigated the problem of estimating the uniform weight $w(X,{\mathcal{U}}_n)$ of a Tychonoff space X equipped with its fine uniformity. Building upon previous results that connect uniform weights with cozero sets and cardinal functions, we have established a variety of upper bounds for $w(X,{\mathcal{U}}_n)$ in terms of the cardinal $z\left(X\right)$ and strong cardinal invariants associated with the diagonal $\Delta \left(X\right)$ in extensions of $X\times X$.

We have shown that under several natural conditions such as pseudocompactness, paracompactness, or $C_1$-embedding, the fine uniform weight can be effectively bounded by functions involving $z\left(X\right)$ and the pseudocharacter of $\Delta \left(X\right)$. Our main result offers a general upper bound that depends on the fine weights of closed subspaces and on the existence of L-local bases of small cardinality.

These findings contribute to the broader classification of uniform spaces by their intrinsic cardinal invariants and clarify the connections between fine uniformities, topological completeness, and function-theoretic structures. Several open questions remain, particularly regarding whether $w(X,{\mathcal{U}}_n)\le z\left(X\right)$ holds in full generality, which invites further exploration in future work.