1. Introduction
As a stronger notion than an accumulation point, a condensation point of a subset
A of a topological space
is every point
x in
X such that every open neighborhood of
x uncountably contains many points of
A. The notions of
-closed sets and
-open sets were introduced by Hdeib [
1]. Those subsets of a topological space containing their condensation points are called
-closed sets, and their complements are called
-open sets. Via
-open sets, the author in [
1] characterized Lindelöf topological spaces; the authors in [
2,
3,
4] introduced some types of continuity. Moreover, the authors in [
5,
6,
7,
8,
9] introduced many modifications of both
-open and
-closed sets. Recently, the authors in [
10,
11,
12,
13,
14,
15,
16] obtained several interesting results related to these sets. The authors in [
17,
18,
19] introduced
-open sets in fuzzy topological spaces and soft topological spaces. The area of research related to
-open sets is still a hot topic in the field topology structure and its modifications. As a new mathematical structure, the notion of a generalized topological space (GTS) was initiated by Császár in [
20]. The pair
where
is a family of subsets of
X, is called a GTS if
contains the empty set and an arbitrary union of members if
is a member of
. For a GTS
, the members of
are called
-open sets and their complements are called
-closed sets, also subsets of
X which are
-open and
-closed are called
-clopen. Since Császár defined GTSs, various authors have modified many topological concepts to include the structure of GTSs. In [
21], the authors introduced the notion of a slightly
-continuous function between GTSs in which they gave some preservation theorems. In a part of this paper, we continue the study of slightly
-continuous functions, in particular, we introduce some characterizations and preservation theorems. In [
22], the definition of
-open sets was extended to include GTSs. The work of [
22] was continued by [
23,
24,
25]. In this paper, we extend the study of slightly
-continuity [
26] to include the structure of GTSs, and we introduce and investigate the concept of a slightly
-irresolute function between GTSs. The study deals with the preservation of Lindelofness, mild Lindelofness, mildly countable compactness, and connectedness under images or inverse images of a class of functions, which gives symmetric relationships. Note that a GTS
contains
-clopen sets only if
. Therefore, in this work, for each GTS
, we will assume that
.
The structure of this paper lies is as follows. In
Section 2, we collect some known results regarding slight continuity, we also introduce some new results regarding slight continuity. In
Section 3, we introduce and investigate slightly
-
-continuous functions. In
Section 4, we introduce and investigate slightly
-
-irresolute functions. In
Section 5, we deal with the preservation of several covering properties under inverse images of a certain class of functions.
Throughout this paper, we follow the notions and terminologies as they appeared in [
22]:
and
denote the rationals and irrationals, respectively, and
denote the usual topology on
.
2. Slightly -Continuous Functions
In 1997, Singal et al. introduced the concept of slight continuity as a generalization of continuity in [
27].
The authors in [
21] extended slight continuity to include GTSs as follows:
Definition 1. [21] A function between the GTSs and is called slightly -continuous if is -open for every -clopen subset . Theorem 1. [21] For any function between the GTSs and , the following statements are equivalent: (a) f is slightly -continuous;
(b) For each -clopen set , is -closed;
(c) For each -clopen set , is -clopen.
Definition 2. Let X be a nonempty set and β be a collection of subsets of X. Then, the collection of all possible unions of elements of β together with ∅ and X will be denoted by .
Proposition 1. Let X be a nonempty set and β be a collection of subsets of X. Then, the collection forms a GT on X.
Each one of the following two bijection functions is not slightly -continuous:
Example 1. Let and
. Define
by Let . Since B is -clopen but is not -open, then f is not slightly -continuous.
Example 2. Let and . Define by Let . Since B is -clopen but is not -open, then f is not slightly -continuous.
Theorem 2. [21] -continuous functions are slightly -continuous. The converse of Theorem 2 need not be true in general as each of the following three examples clarifies:
Example 3. Let and . Define by The -clopen sets of X are ∅, X, A, and B. In addition, and . This shows that f is slightly -continuous. On the other hand, since but , then f is not -continuous.
Example 4. Let , , , and . Define by . Then, f is slightly -continuous but not -continuous.
Example 5. Let , . Consider the Dirichlet function by if x is rational and if x is irrational. Then, f is slightly -continuous but not -continuous.
Definition 3. A subset H of a GTS is called -μ-open if for each , there is a μ-clopen set M of such that . Complements of -μ-open sets are called -μ-closed.
Proposition 2. Let H be a subset of a GTS . Then,
(a) If H is μ-clopen, then H is -μ-open;
(b) If H is μ-clopen, then H is -μ-closed.
Proof. (a) For each , we have and H is -clopen. This shows that H is --open;
(b) Since H is -clopen, then is -clopen and by (a), is --open. Hence, H is --closed. □
The following example shows that the converse of each implication of Proposition 2 need not be true in general:
Example 6. Let , , , , , and . Then, is --open but not -clopen since is not -open. This shows that the converse of Proposition 2 (a) is not true in general. On the other hand, the set is a --closed that is not -clopen, which shows that the converse of Proposition 2 (b) is not true in general.
Theorem 3. For any function between the GTSs and , the following statements are equivalent:
(a) f is slightly -continuous;
(b) is -open for every --open set ;
(c) is -open for every --closed set .
Proof. (a) ⟹ (b): Let
H be a
-
-open subset of
Y. For each
, choose a
-clopen subset
such that
. By Theorem 1, for each
,
is
-open. We have
It follows that is -open.
(b) ⟹ (c): Let H be a --closed subset of Y. Then, H is --open and by (b), is -open. Hence, is -closed.
(c) ⟹ (a): Follows from Proposition 2 (b) and part (b) of Theorem 1. □
Definition 4. [28] A GTS is called μ-connected if we cannot find such that and . Theorem 4. Let be a function between the GTSs and . If is -connected, then f is slightly -continuous.
Remark 1. The GTS in Example 3 is not -connected as are nonempty disjoint sets and . Therefore, the implication in Theorem 4 is not reversible.
Theorem 5. [21] Let be a surjective -continuous function between the GTSs and . If is -connected, then is -connected. Definition 5. [21] A GTS is said to be μ-zero-dimensional if there is such that each element is μ-clopen and . Theorem 6. [21] Let be a function between the GTSs and , where is -zero-dimensional. Then, the following statements are equivalent: (a) f is -continuous;
(b) f is slightly -continuous.
Remark 2. Example 4 shows that the condition “-zero-dimensional” on the GTS cannot be dropped.
Definition 6. [29] Let be a GTS and H a nonempty subset of Y. The relative GT of H on Y is the GT on H. The pair is called a relative GTS of . Theorem 7. Let be a slightly -continuous function between the GTSs and and let . Then, the restriction of f to H, is slightly -continuous.
Proof. Let
B be a
-clopen subset of
Y. Since
f is slightly
-continuous,
is
-open. Thus,
Hence, is slightly -continuous. □
Definition 7. [30] Let and be two GTSs and let We call the product of the GTSs μ and σ and denote it by . The GTS is called the product of the GTSs and .
Definition 8. [20] Let be a GTS and H a subset of X. Then, (a) The μ-closure of H; denoted by ; is defined by (b) The μ-interior of H; denoted by ; is defined by Proposition 3. [30] Let and be two GTSs, and . Then, (a) ;
(b) .
Corollary 1. Let and be two GTSs. If is μ-clopen and is σ-clopen, then is -clopen.
Definition 9. [31] A GTS is called μ- if for all with , there are μ-open sets such that , and . Definition 10. A GTS is called μ-clopen- if for all with , there are μ-clopen sets such that , and .
Theorem 8. If is a μ-clopen- GTS, where X contains at least two points, then is not μ-connected.
Proof. Suppose to the contrary that is -connected. Choose with . Since is -clopen-, there are -clopen sets such that , , and . Thus, A and are -open nonempty disjoint sets with . □
Theorem 9. μ-clopen- GTSs are μ-.
Remark 3. Since is - and -connected, then by Theorem 8, is not -clopen-.
Theorem 10. If the GTS is σ- and σ-zero-dimensional, then is σ-clopen-.
Proof. By -zero-dimensionality of , there is such that each element is -clopen and . To see that is -clopen-, let with . Since is -, there are -open sets such that , and . Choose such that and . Then, and H are -clopen sets such that , and . This shows that is -clopen-. □
In each of the following two examples, the GTS is -clopen- but is not a topological space:
Example 7. Let and . Then, is a -clopen- GTS, but is not a topological space.
Example 8. Let and . Then, is a -clopen- GTS, but is not a topological space.
Theorem 11. If is a slightly -continuous function, where is σ-clopen-, then the graph of f is -closed.
Proof. Let . Then, . As is -clopen-, we find -clopen sets such that and and . Since f is slightly -continuous, . Now, and . Therefore, , and hence is -closed. □
Theorem 12. Let be a GTS and be a σ-clopen- GTS. If is a slightly -continuous function, then the set is a -closed subset of .
Proof. Put . Let . Then, . Since is -clopen-, there exist disjoint -clopen sets such that and . Since f is slightly -continuous, then . We have and . This proves that , and hence H is -closed. □
3. Slightly --Continuous Functions
As a weaker form of continuity between GTSs, the authors in [
22] defined
-continuity between GTSs as follows:
Definition 11. A function between the GTSs and is called ω--continuous if for each -open subset , is ω--open.
Definition 12. A function between the GTSs and is called slightly ω--continuous if for each -clopen subset , is ω--open.
Example 9. Let be as in Example 7. Define by Then, g is slightly ω--continuous.
Proof. Note that the set of all
-clopen subsets of
is
together with ∅ and
. Let
. Then,
and
Therefore, for all . It follows that g is slightly --continuous. □
Example 10. Consider the function f as in Example 2. Then,
(a) ;
(b) f is not slightly ω--continuous.
Proof. (a) Suppose to the contrary that there are such that and . Then, there are and a countable subset such that . Choose such that . Then, we have , and so is countable, which is a contradiction;
(b) is -clopen, but by (a) . □
Theorem 13. ω--continuity between the GTSs and implies slight ω--continuity.
Proof. The proof follows from Proposition 3.4 of [
22] and the fact that
-clopen sets are
-open. □
The implication in Theorem 13 is not reversible:
Example 11. Let and . Consider , where Since is a δ-open set but is not ω-σ-open, then g is not ω--continuous. On the other hand, since the only δ-clopen subsets are ∅ and , then g is slightly ω--continuous.
Theorem 14. Slight -continuity between the GTSs and implies slight ω--continuity.
Proof. Th proof follows from the definitions and the fact that -open sets are --open sets. □
The following example shows that the converse of Theorem 14 in not true in general:
Example 12. Let and . Define by Let . Since B is -clopen but , then f is not slightly -continuous. On the other hand, it is easy to check that is the discrete topology on X and hence f is slightly ω--continuous.
Theorem 15. If is a function between the GTSs and , where is σ-connected, then f is slightly ω--continuous.
Theorem 16. For any function between the GTSs and , the following statements are equivalent:
(a) f is slightly ω--continuous;
(b) is ω--closed for every -clopen set ;
(c) is ω--clopen for every -clopen set .
Proof. Straightforward. □
The converse of Theorem 13 is true if we add the condition ‘-zero-dimensional’ on the GTS as the following result shows:
Theorem 17. If is a slightly ω--continuous function between the GTSs and , where is σ-zero-dimensional, then f is ω--continuous.
Proof. Let
. Since
is
-zero-dimensional, there is
such that each element
is
-clopen, and
. Choose
such that
. By the slight
-
-continuity of
f,
for all
. Since
then
is
-
-open. □
As defined in [
22], a GTS
is said to be
-anti-locally countable if the intersection of any two
-open sets is either empty or uncountable.
Proposition 4. Let be σ-anti-locally countable and . If , then .
Proof. We may assume . It is clear that . To see that , let and let such that . Choose and a countable set such that . Since , . Choose . Since , there exists and a countable set such that . Since and is -anti-locally countable, then is uncountable. Thus, and hence . Therefore, . □
Definition 13. Let be a GTS and . Then, H is said to be ω-σ-clopen if it is both ω-σ-open and ω-σ-closed.
Theorem 18. Let be σ-anti-locally countable and . Then, H is σ-clopen if H is ω-σ-clopen.
Proof. ⟹) Suppose that H is -clopen, then H and are -open. Thus, H and are --open, and hence H is --clopen.
⟸) Suppose that
H is
-
-clopen. Since
H and
are
-
-open sets, then by Proposition 4
Since
H is
-
-closed, then
Therefore,
and hence
H and
are
-closed sets. It follows that
H is
-clopen. □
Theorem 19. If is a slightly ω--continuous function between the GTSs and , where is -anti-locally countable, then f is slightly -continuous.
Proof. Let B be -clopen. By slight --continuity of f and part (c) of Theorem 16, is --clopen. Since is -anti-locally countable, is -clopen. Therefore, by part (c) of Theorem 1, we get the result. □
Theorem 20. For any function between the GTSs and , the following statements are equivalent:
(a) f is slightly ω--continuous;
(b) is ω--open for every --open set ;
(c) is ω--closed for every --closed set .
(a) ⟹ (b): Let B be a --open subset of Y. For each , choose a -clopen subset such that . For each , is ω--open. We have It follows that is ω--open.
(b) ⟹ (c): Let B be a --closed subset of Y. Then, is --open and by (b), is ω--open. Hence, is ω--closed.
(c) ⟹ (a): Follows from Proposition 2 (b) and part (b) of Theorem 16.
Definition 14. A GTS is called ω-μ-connected if there are no nonempty disjoint ω-μ-open sets such that .
Theorem 21. If is an ω-μ-connected GTS, then is μ-connected.
Proof. By contraposition. If is not -connected, then there exist such that and . Thus, we have as nonempty disjoint --open sets with which shows is not --connected. □
The implication in the above theorem is not reversible as the following two examples show:
Example 13. Let . Then, the GTS is
(a) μ-connected;
(b) not ω-μ-connected.
Proof. (a) Suppose to the contrary that is not -connected. Then, there exist nonempty disjoint -open sets A, B with . Since is -connected, then or . Without loss of generality, we may assume that . Then, there is such that . So, . However, the only -open set contained in is ∅, and so which is a contradiction.
(b) For every , ; because with and is a countable subset of ; and . Put and , then are nonempty disjoint -open sets with . This implies that is not --connected. □
Example 14. Let and . Then, is a μ-connected GTS. On the other hand, set and . Then, are nonempty disjoint ω-μ-open sets with . This implies that is not ω-μ-connected.
Theorem 22. Let be a μ-anti-locally countable GTS. Then, the following are equivalent:
(a) is ω-μ-connected;
(b) is μ-connected.
Proof. (a) ⟹ (b): Follows from Theorem 21.
(b) ⟹ (a): By contraposition. Suppose is not --connected. Then, there are nonempty disjoint --open sets such that . It is not difficult to see that A and B are --clopen. Thus, by Theorem 18, A and B are -clopen. Therefore, is not -connected. □
Corollary 2. is ω--connected.
Proof. It is not difficult to check that is -anti-locally countable. Since is -connected, then by Theorem 22, is --connected. □
The following is a natural question:
Is there an --connected GTS where is not a topology on X?
The answer is yes, as the following example shows:
Example 15. Let . Then, is a GTS that is not a topological space. It is not difficult to check that is -anti-locally countable and -connected. Therefore, by Theorem 22, is ω--connected.
Theorem 23. Let be a surjective slightly ω--continuous function between the GTSs and , where is ω--connected, then is -connected.
Proof. Suppose to the contrary that
is not
-connected. Then, there are
such that
and
. Since it is clear that
are
-clopen subsets of
Y and
f is slightly
-
-continuous, then
are
-
-open sets. Since
f is surjective and
are nonempty, then
are nonempty. In addition,
and,
It follows that is not --connected, which is a contradiction. □
4. Slightly --Irresolute Functions
Definition 15. A function between the GTSs and is called ω--irresolute if for each ω-σ-open subset B of Y, the inverse image is an ω-μ-open subset of X.
Definition 16. A function between the GTSs and is called slightly ω--irresolute if for each ω-σ-clopen subset B of Y, the inverse image is an ω-μ-open subset of X.
Example 16. Let , where is the natural numbers. Define byand Then,
(a) f is slightly ω--irresolute;
(b) g is not slightly ω--irresolute.
Proof. (a) It is not difficult to check that the set of all --clopen subsets of is . If , then and so . If , then when and when , and so . It follows that f is slightly --irresolute.
(b) Since is --clopen but is not --open, then g is not slightly --irresolute. □
Theorem 24. Let be a function between the GTSs and .
(a) If f is ω--irresolute, then f is slightly ω--irresolute;
(b) If f is slightly ω--irresolute, then f is slightly ω--continuous;
(c) If is ω-σ-connected, then f is slightly ω--irresolute;
(d) If f is ω--irresolute, then f is ω--continuous.
Proof. (a) Suppose that f is --irresolute and let B be an --clopen subset of Y. Then, B is --open. Since f is --irresolute, then is an --open subset of X. Therefore, f is slightly --irresolute;
(b) Suppose that f is slightly --irresolute and let B be a -clopen subset of Y. Then, B is --clopen. Since f is slightly --irresolute, then is an --open subset of X. Therefore, f is slightly --continuous;
(c) and (d) are obvious. □
The converse of Theorem 24 (a) is not true in general as the following example shows:
Example 17. Let and . Define by By Corollary 2, is ω-σ-connected and so f is obviously slightly ω--irresolute. On the other hand, since is ω-σ-open but , then f is not ω--irresolute.
The converse of Theorem 24 (b) is not true in general as the following example shows:
Example 18. Consider the function as in Example 11. Let the discrete topology on , = the co-countable topology on and the family of δ-clopen subsets of . It is easy to check that . Thus, . Since is ω-δ-clopen but is not ω-σ-open, then g is not slightly ω--irresolute. On the other hand, we proved in Example 11 that g is slightly ω--continuous.
Theorem 25. Let be a function between the GTSs and , and let be a function between the GTSs and .
(a) is slightly ω--irresolute if f is slightly ω--irresolute and g is slightly ω--irresolute;
(b) is slightly ω--continuous if f is slightly ω--irresolute and g is slightly ω--continuous;
(c) is slightly ω--continuous if f is slightly ω--continuous and g is slightly -continuous.
Proof. (a) Let
C be an
-
-clopen subset of
Z. Since
g is slightly
-
-irresolute, then
is
-
-clopen. Since
f is slightly
-
-irresolute, then
is
-
-open subset of
X. Since
then
is
-
-open set in
X. Hence,
is slightly
-
-irresolute;
(b) Let
C be a
-clopen subset of
Z. Since
g is slightly
-
-continuous, then
is an
-
-clopen subset of
Y. Since
f is slightly
-
-irresolute, then
is
-
-open subset of
X. Since
then
is
-
-open set in
X. Hence,
is slightly
-
-continuous;
(c) Let
C be a
-clopen subset of
Z. Since
g is slightly
-continuous, then
is
-clopen. Since
f is slightly
-
-continuous, then
is
-
-open subset of
X. Since
then
is
-
-open set in
X. Hence,
is slightly
-
-continuous. □
Corollary 3. If is an ω--irresolute function between the GTSs and and is a slightly ω--continuous function between the GTSs and , then is slightly ω--continuous function.
Definition 17. A function is called ω--open function if for every ω-μ-open subset H of X, is ω-σ-open in Y.
Theorem 26. Let be slightly ω--irresolute, ω--open, and surjective function between the GTSs and , and let be a function between the GTSs and . Then, the following are equivalent:
(a) g is slightly ω--continuous;
(b) is slightly ω--continuous.
proof. (a) ⟹ (b): Theorem 25 (b).
(b) ⟹ (a): Let C be a η-clopen subset of Z. Since is slightly ω--continuous, is ω-μ-open. Since f is ω--open, is ω-σ-open. Since f is surjective, then . It follows that is ω-σ-open.
5. Covering Properties
Definition 18. [32] A subset H of a GTS is called σ-Lindelof relative to Y if every cover of H by σ-open sets has a countable subcover. A GTS is called σ-Lindelof if each cover of Y by σ-open sets has a countable subcover. Remark 4. A GTS is σ-Lindelof if Y is σ-Lindelof relative to Y.
Definition 19. A function between the GTSs and is called -closed if is -closed for each -closed set C.
Recall that for a function and for , the set is called a fiber of g.
Definition 20. A function between the GTSs and is called -perfect if g is -continuous, -closed, and its fiber is -compact relative to Y for all .
Definition 21. A subset H of a GTS is called σ-mildly-compact relative to Y if every cover of H by σ-clopen sets has a finite subcover.
Definition 22. [21] A GTS is called σ-mildly compact if each cover of Y by σ-clopen sets has a finite subcover. Remark 5. A GTS is σ-mildly-compact if Y is σ-mildly-compact relative to Y.
Definition 23. A function between the GTSs and is called strongly -closed if is -clopen for all -closed set C.
Theorem 27. Let be strongly -closed between the GTSs and in which its fibers are -mildly-compact relative to Y. If B is -mildly-compact relative to Z, then is -mildly-compact relative to Y.
Proof. Let
be a cover of
by
-clopen sets. Let
be the family of all finite subsets of
. For all
, set
. For each
,
is
-mildly-compact relative to
Y and so there exists
such that
. Set
□
Claim. is a cover of B by -clopen sets.
Proof of Claim. Let . Then, and so with . It follows that is a cover of B. Furthermore, for every , is -open, is -closed, and since g is strongly -closed, is -clopen. It follows that is a cover of B by -clopen sets.
Since
B is
-mildly-compact relative to
Z and by the Claim,
is a cover of
B by
-clopen sets, there are
such that
Put . Then, is a finite subcover of . Therefore, is -mildly-compact relative to Y. □
Corollary 4. Let be strongly -closed between the GTSs and in which its fibers are -mildly-compact relative to Y. If is -mildly-compact, then is -mildly-compact.
Definition 24. A subset H of a GTS is called σ-mildly-countably-compact relative to Y if every countable cover of H by σ-clopen sets has a finite subcover.
Definition 25. A GTS is called σ-mildly-countably-compact if each countable cover of Y by σ-clopen sets has a finite subcover.
Remark 6. A GTS is σ-mildly-countably-compact if Y is σ-mildly-countably-compact relative to Y.
Theorem 28. Let be strongly -closed between the GTSs and in which its fibers are -mildly-countably-compact relative to Y. If B is -mildly-countably-compact relative to Z, then is -mildly-countably-compact relative to Y.
Proof. Let
be a countable cover of
by
-clopen sets. Let
be the family of all finite subsets of
. For all
, set
. For each
,
is
-mildly-countably-compact relative to
Y and so there exists
such that
. Set
□
Claim. is a countable cover of B by -clopen sets.
Proof of Claim. Let . Then, and so with . It follows that is a cover of B. In addition, for every , is -open, is -closed, and since g is strongly -closed, is -clopen. It follows that is a countable cover of B by -clopen sets.
Since
B is
-mildly-countably-compact relative to
Z, and by the Claim,
is a cover of
B by
-clopen sets, there are
such that
Put . Then, is a finite subcover of . Therefore, is -mildly-countably-compact relative to Y. □
Corollary 5. Let be strongly -closed between the GTSs and in which its fibers are -mildly-countably-compact relative to Y. If is -mildly-countably-compact, then is -mildly-countably-compact.
Theorem 29. Let be -closed between the GTSs and in which its fibers are -Lindelof relative to Y. If B is -Lindelof relative to Z, then is -Lindelof relative to Y.
Proof. Let
be a cover
by
-open sets. Let
be the family of all countable subsets of
. For all
, set
. For each
is
-Lindelof relative to
Y and so there exists
such that
. Set
□
Claim. is a cover of B by -open sets.
Proof of Claim. Let . Then, and so with . It follows that is a cover of B. Additionally, for every , is -open, is -closed, and since g is -closed, is -closed. It follows that is a cover of B by -open sets.
Since
B is
-Lindelof relative to
Z, and by the Claim,
is a cover of
B by
-open sets, there exists a countable subfamily
such that
Put . Then, is a countable subcover of . Therefore, is -Lindelof relative to Y. □
Corollary 6. Let be -perfect between the GTSs and . If B is -Lindelof relative to Z, then is -Lindelof relative to Y.
Corollary 7. Let be -closed between the GTSs and in which its fibers are are -Lindelof relative to Y. If is -Lindelof, then is -Lindelof.
Corollary 8. Let be -perfect between the GTSs and . If is -Lindelof, then is -Lindelof.
Definition 26. A subset H of a GTS is called σ-mildly-Lindelof relative to Y if every cover of H by σ-clopen sets has a countable subcover.
Definition 27. A GTS is called σ-mildly-Lindelof if each cover of Y by σ-clopen sets has a countable subcover.
Remark 7. A GTS is σ-mildly-Lindelof if Y is σ-mildly-Lindelof relative to Y.
Theorem 30. Let be strongly -closed between the GTSs and in which its fibers are -mildly-Lindelof relative to Y. If B is -mildly-Lindelof relative to Z, then is -mildly-Lindelof relative to Y.
Proof. Let
be a cover of
by
-clopen sets. Let
be the family of all countable subsets of
. For all
, set
. For each
,
is
-mildly-Lindelof relative to
Y and so there exists
such that
. Set
□
Claim. is a cover of B by -clopen sets.
Proof of Claim. Let . Then, and so with . It follows that is a cover of B. Furthermore, for every , is -open, is -closed, and since g is strongly -closed, is -clopen. It follows that is a cover of B by -clopen sets.
Since
B is
-mildly-Lindelof relative to
Z, and by the Claim,
is a cover of
B by
-clopen sets, there is a countable subfamily
such that
Put . Then, is a countable subcover of . Therefore, is -mildly-Lindelof relative to Y. □
Corollary 9. Let be strongly -closed between the GTSs and in which its fibers are -mildly-Lindelof relative to Y. If is -mildly-Lindelof, then is -mildly-Lindelof.