MATEMATIQKI VESNIK

59 (2007), 23–30

UDK 515.122

originalni nauqni rad research paper

ON COLC TOPOLOGIES

A. Kanibir and P. Girginok

Abstract. In this paper we introduce the concept of coLC topologies and discuss some of their basic properties. We relate this concept to classes of functions between topological spaces.

1. Introduction

A topological space whose Lindelöf subsets are closed is called an LC-space by Mukherji and Sarkar [17] and by Gauld, Mršević, Reilly and Vamanamurthy [7]. LC-spaces are also known as L-closed spaces [8,9,10,15]. They specialize KCspaces, i.e. spaces in which compact subsets are closed [18] and Hausdorff P-spaces, i.e. spaces in which F_{σ}-sets are closed [16]. LC-spaces have been of some interest during recent years [1,2,3,4].

In 1984, Gauld, Mršević, Reilly and Vamanamurthy [7] defined the coLindelöf topology of τ on X. Let (X, τ) be a topological space. Then ℓ(τ) = {∅} ∪ {G ∈ τ : X − G is Lindelöf in (X, τ)} is a topology on X with ℓ(τ) ⊆ τ, and it is called the coLindelöf topology of τ on X. This topology is analogous to the cocompact topology c(τ) which is considered by Gauld [5] and [6] and by Kohli [11].

The purpose of the present paper is to introduce the concept of the coLC topology of τ on X denoted by ℓc(τ) and to study its basic properties. In Section 2, by giving an example, it is observed that X with the coLC topology of τ on X is not necessarily an LC-space. We investigate the conditions under which the topological space (X, ℓc(τ)) is an LC-space where (X, τ) is a topological space. We relate ℓc(τ) to the cocompact topology of (X, τ) and to the coLindelöf topology ℓ(τ) of (X, τ). We also investigate relationship between the coLC topology of the subspace topology and the subspace topology of ℓc(τ) for any subset of X. In the last section we give some continuity results. Our terminology is standard. The closure and interior of a subset Y of a space (X, τ) are denoted by τclY and τintY, respectively. Denote the relative topology on Y by τ_{Y}. Given a topological space (X, τ) and a subset Y ⊆ X, ℓc(τ_{Y}) and ℓc(τ)_{Y} denote the coLC topology of the relative topology τ_{Y} on X and the relative topology induced by the coLC topology ℓc(τ) of τ on X, respectively.

AMS Subject Classification: 54A05, 54D20, 54G10

Keywords and phrases: Lindelöf, LC-space, coLindelöf topology, coLC topology.

2. CoLC topologies

Firstly we give the following results which will be needed in the sequel.

Proposition 1. The union of any two closed LC-subspaces of (X, τ) is an LC-subspace.

Proof. Let K_{1}, K_{2} ⊆ X be closed, LC-subspaces of (X, τ). Let K be a Lindelöf subset of (K_{1} ∪ K_{2}, τ_{K_{1}∪K_{2}}). Then K ∩ K_{1} is a Lindelöf subset of (K_{1}, τ_{K_{1}}). Since K_{1} is an LC-subspace of (X, τ), K ∩ K_{1} is closed in (K_{1}, τ_{K_{1}}). The set K ∩ K_{1} is closed in X since K_{1} is closed in (X, τ). Similarly K ∩ K_{2} is closed in (X, τ). Thus (K ∩ K_{1}) ∪ (K ∩ K_{2}) = K is closed in (X, τ). This implies that K is closed in (K_{1} ∪ K_{2}, τ_{K_{1}∪K_{2}}). Hence K_{1} ∪ K_{2} is an LC-subspace of (X, τ).

Corollary 2. Let X be a topological space. If X can be expressed as the union of two closed LC-subspaces, then X is an LC-space.

Definition 3. Let (X, τ) be a topological space. The collection ℓc(τ) = {∅} ∪ {U ∈ τ : X − U is an LC-subspace in (X, τ)} is a topology on X with ℓc(τ) ⊆ τ, called the coLC topology of τ on X.

It is easy to see that ℓc(τ) is a topology on X since the LC property is hereditary [17] and by Proposition 1.

The next example shows that (X, ℓc(τ)) is not necessarily an LC-space, even if (X, τ) is Lindelöf.

Example 4. Let X be an uncountable set and τ be the countable complement topology on X. The coLC topology of τ on X is τ, that is τ = ℓc(τ). Since (X, τ) is not an LC-space [17], (X, ℓc(τ)) is not an LC-space.

Proposition 5. If (X, τ) is an LC-space, then τ = ℓc(τ).

Proof. By definition ℓc(τ) ⊆ τ. Now take U ∈ τ. Then X − U is an LCsubspace of (X, τ) since X is an LC-space and the LC property is hereditary. Hence U ∈ ℓc(τ).

Example 4 shows that the reverse implication of Proposition 5 does not hold in general.

Proposition 6. Let (X, τ) be a topological space. If τ = ℓc(τ) and (X, τ) has no Lindelöf-dense subset, then (X, τ) is an LC-space.

Proof. Let L be a Lindelöf subset of (X, τ). Then L is Lindelöf in the subspace τclL of (X, τ). Now X − τclL ≠ ∅ by hypothesis. Since X − τclL ∈ τ = ℓc(τ) we have that X − (X − τclL) = τclL is an LC-subspace of (X, τ) and hence L is closed in τclL. Since τclL is closed in (X, τ), L is closed in (X, τ). Hence (X, τ) is an LC-space.

In the above theorem the condition that (X, τ) has no Lindelöf-dense subset can not be removed as the Example 4 shows.

On coLC topologies 25

Corollary 7. Let (X, τ) be a topological space. If (X, ℓc(τ)) has no Lindelöfdense subset then (X, ℓc(τ)) is an LC-space.

Proof. Firstly we show that (X, τ) is an LC-space. Let L be a Lindelöf subset of (X, τ). Then L is Lindelöf in the subspace τclL of (X, τ). We have τclL ⊆ ℓc(τ)clL and ℓc(τ)clL ≠ X by hypothesis. Since X − ℓc(τ)clL ∈ ℓc(τ), ℓc(τ)clL is an LCsubspace of (X, τ). Then τclL is an LC-subspace of (X, τ). Hence L is closed in τclL. Since τclL is closed in (X, τ), L is closed in (X, τ). Thus (X, τ) is an LC-space. We obtain τ = ℓc(τ) from Proposition 5 and hence (X, ℓc(τ)) is an LC-space.

Theorem 8. For a space (X, τ) the following are equivalent:

(1) (X, τ) is an LC-space.

(2) (X, ℓc(τ)) is an LC-space.

Proof. (1)⇒(2) It follows from Proposition 5.

(2)⇒(1) Let L be a Lindelöf subset of (X, τ). All Lindelöf subsets of (X, τ) are Lindelöf in (X, ℓc(τ)) since ℓc(τ) ⊆ τ. Thus L is Lindelöf in (X, ℓc(τ)) and so L is closed in (X, ℓc(τ)) since (X, ℓc(τ)) is an LC-space. Hence L is closed in (X, τ).

Remark 9. Let (X, τ) be a topological space which is not countable discrete. If (X, ℓc(τ)) is a separable space, then (X,ℓc(τ)) and (X, τ) are not LC-spaces. Hence if (X, ℓc(τ)) is second countable, then (X, ℓc(τ)) and (X, τ) are not LCspaces.

Proposition 10. If (X, ℓc(τ)) is Hausdorff, then (X, τ) is an LC-space.

Proof. Let x, y ∈ X and x ≠ y. Since (X, ℓc(τ)) is Hausdorff, there are U, V ∈ ℓc(τ) such that x ∈ U, y ∈ V and U ∩ V = ∅. Thus X = (X − U) ∪ (X − V) and hence X can be written as the union of two closed, LC-subspaces of (X, τ). By Corollary 2 (X, τ) is an LC-space.

Corollary 11. Let (X, τ) be a topological space. If (X, ℓc(τ)) is Hausdorff, then (X, ℓc(τ)) is an LC-space.

The following example shows that if (X, τ) is Hausdorff, then (X, τ) is not necessarily an LC-space.

Example 12. Let R be the set of real numbers and τ be the usual topology on R. Then (R, τ) is Hausdorff. The subset Q of rational numbers is Lindelöf but not closed. Hence (R, τ) is not an LC-space.

Proposition 13. Let (X, τ) be a topological space. (X, ℓc(τ)) is an LC-space if it can be expressed as the union of two closed sets which are not equal to X.

Proof. Let X = K_{1} ∪ K_{2} and let K_{1} ≠ X, K_{2} ≠ X be closed in (X, ℓc(τ)). Then K_{1} and K_{2} are LC-subspaces of (X, τ) by Definition 3. Therefore (X, τ) is an LC-space by Corollary 2, since K_{1} and K_{2} are closed in (X, τ). Hence (X, ℓc(τ)) is an LC-space by Theorem 8.

Corollary 14. If (X, ℓc(τ)) is not connected, then (X, ℓc(τ)) is an LC-space.

Theorem 15. Let (X, τ) be a topological space. Then (X, τ) is an LC-space and not connected if and only if (X, ℓc(τ)) is not connected.

Proof. (⇒) Let (X, τ) be an LC-space and non-connected. Then τ = ℓc(τ) by Proposition 5. Hence (X, ℓc(τ)) is not connected.

(⇐) (X, τ) is not connected since (X, ℓc(τ)) is non-connected and ℓc(τ) ⊆ τ. By Corollary 14, (X, ℓc(τ)) is an LC-space since (X, ℓc(τ)) is non-connected. Hence (X, τ) is an LC-space by Theorem 8.

The following four generalizations of LC-spaces and Theorem are given by Dontchev, Ganster and Kanibir [2].

Definition 16. A topological space (X, τ) is called:

(1) an L_{1}-space if every Lindelöf F_{σ}-set is closed,

(2) an L_{2}-space if τclL is Lindelöf whenever L ⊆ X is Lindelöf,

(3) an L_{3}-space if every Lindelöf subset L is an F_{σ}-set,

(4) an L_{4}-space if, whenever L ⊆ X is Lindelöf, then there is a Lindelöf F_{σ}-set F with L ⊆ F ⊆ τclL.

Theorem 17. (1) If (X, τ) is an LC-space, then (X, τ) is an L_{i}-space, i = 1, 2, 3, 4.

(2) Every L_{2}-space is an L_{4}-space and every L_{3}-space is an L_{4}-space.

Proposition 18. Let (X, τ) be a topological space. If (X, ℓc(τ)) has no Lindelöf-dense subset, then (X, τ) and (X, ℓc(τ)) are L_{i}-spaces, i = 1, 2, 3, 4.

Proof. (X, ℓc(τ)) is an LC-space by Corollary 7 since (X, ℓc(τ)) has no Lindelöf-dense set. Thus (X, τ) is an LC-space. By Theorem 17(1) (X, τ) and (X, ℓc(τ)) are L_{i}-spaces, i = 1, 2, 3, 4.

Proposition 19. Let (X, τ) be a topological space. If (X, ℓc(τ)) is an L_{2}-space but not an LC-space, then (X, ℓc(τ)) is a Lindelöf space.

Proof. Let (X, ℓc(τ)) be L_{2} and not an LC-space. Then there is a Lindelöfdense set L in (X, ℓc(τ)) by Corollary 7 and ℓc(τ)clL = X. Hence (X, ℓc(τ)) is a Lindelöf space since (X, ℓc(τ)) is an L_{2}-space.

A space (X, τ) satisfying the hypothesis of the previous Proposition is given by Example 4.

Proposition 20. If (X, τ) is T_{1}, then (X, ℓc(τ)) is T_{1}.

Proof. Let x ∈ X. Since (X, τ) is T_{1} and {x} is an LC-subspace of (X, τ), X − {x} ∈ ℓc(τ). Hence {x} is closed in (X, ℓc(τ)). Thus (X, ℓc(τ)) is T_{1}.

Example 12 shows that if (X, τ) is Hausdorff then (X, ℓc(τ)) is not necessarily Hausdorff. (R, τ) is Hausdorff, but is not an LC-space. Hence (R, ℓc(τ)) is not Hausdorff by Proposition 10.

Recall that if (X, τ) is a topological space, then the cocompact topology of τ on X is c(τ) = {∅} ∪ {U ∈ τ : X − U is compact in (X, τ)} [5].

Proposition 21. Let (X, τ) be a topological space.

(1) If (X, τ) is an LC-space, then ℓ(τ) ⊆ ℓc(τ) and c(τ) ⊆ ℓc(τ).

(2) If (X, τ) is a Lindelöf space, then ℓc(τ) ⊆ ℓ(τ).

(3) If (X, τ) is a compact space, then ℓc(τ) ⊆ c(τ).

Proof. (1) Let (X, τ) be an LC-space. Then τ = ℓc(τ) by Proposition 4. Since ℓ(τ) ⊆ τ and c(τ) ⊆ τ, ℓ(τ) ⊆ ℓc(τ) and c(τ) ⊆ ℓc(τ). This completes the proof.

(2) Let (X, τ) be a Lindelöf space. Then τ = ℓ(τ) by Corollary 1 of [7]. Since ℓc(τ) ⊆ τ, ℓc(τ) ⊆ ℓ(τ). This completes the proof.

(3) Let (X, τ) be a compact space. Then c(τ) = τ by Corollary 3 of [5]. Hence ℓc(τ) ⊆ τ.

Note that the reverse inclusions of (1), (2) and (3) are false in general.

Example 22. Let X be an uncountable set and let τ be the discrete topology on X. Since X is an LC-space, τ = ℓc(τ) by Proposition 5. It is clear that τ ≠ ℓ(τ) and τ ≠ c(τ). Hence ℓc(τ) ⊈ ℓ(τ) and ℓc(τ) ⊈ c(τ).

Example 23. Let X be the set of real numbers with the usual topology, Y = \{0\} ∪ \{\frac{1}{n} : n = 1, 2, 3, … \} and let τ be the topology induced on the set Y by the topology of X. Since (Y, τ) is compact, it is Lindelöf. Hence τ = c(τ) = ℓ(τ). We take \{\frac{1}{2}\} ∈ τ and A = \{1, \frac{1}{3}, \frac{1}{4}, … \}. Then A ⊆ K = Y − \{\frac{1}{2}\} is a Lindelöf subset of (K, τ_{K}), but not closed in (K, τ_{K}). Therefore K is not an LC-subspace of (Y, τ). So \{\frac{1}{2}\} ∉ ℓc(τ). Hence c(τ) ⊈ ℓc(τ) and ℓ(τ) ⊈ ℓc(τ).

Proposition 24. Let L be a subset of the topological space (X, τ). Then ℓc(τ)_{L} ⊆ ℓc(τ_{L}).

Proof. Suppose U ∈ ℓc(τ)_{L}. Then there is a subset A ∈ ℓc(τ) with U = A ∩ L. Since A ∈ ℓc(τ), X−A is an LC-subspace of (X, τ). Thus L−U is an LC-subspace of (X −A, τ_{X−A}). Hence L−U is an LC-subspace of (X, τ) since (τ_{X−A})_{L−U} = τ_{L−U}. Further U ∈ τ_{L}, so U ∈ ℓc(τ_{L}). Hence ℓc(τ)_{L} ⊆ ℓc(τ_{L}).

Theorem 25. Let (X, τ) be a topological space and Y ⊆ X. If τ = ℓc(τ), then τ_{Y} = ℓc(τ_{Y}) = ℓc(τ)_{Y}.

Proof. Firstly we will show that τ_{Y} ⊆ ℓc(τ_{Y}). Let U_{Y} ∈ τ_{Y} and U_{Y} = U ∩ Y, U ∈ τ. Since τ = ℓc(τ), X − U is an LC-subspace of (X, τ). Therefore Y − U_{Y} is an LC-subspace of (X, τ) because of Y − U_{Y} ⊆ X − U. Since τ_{Y −U_{Y}} = (τ_{Y})_{Y −U_{Y}}, Y − U_{Y} is an LC-subspace of (Y, τ_{Y}). Hence τ_{Y} ⊆ ℓc(τ_{Y}).

We observe that ℓc(τ_{Y}) ⊆ τ_{Y}. Thus τ_{Y} = ℓc(τ_{Y}). We have τ_{Y} = ℓc(τ)_{Y} from the hypothesis. Hence τ_{Y} = ℓc(τ_{Y}) = ℓc(τ)_{Y}.

Note that, if (X, τ) is an LC-space, then Proposition 5 and Theorem 25 imply that τ_{Y} = ℓc(τ_{Y}) = ℓc(τ)_{Y} for every Y ⊆ X.

Note that the τ = ℓc(τ) condition in Theorem 25 can not be removed as the following example shows. This example also shows that the reverse inclusion of Proposition 24 is false in general.

Example 26. Let X = {a, b, c, d}, Y = {a, b} and τ = {∅, X, {a}, {b, c, d}}. Then ℓc(τ) = {∅} ∪ {U ∈ τ : X − U is an LC-subspace in (X, τ)} = {∅, X, {b, c, d}} and τ_{Y} = {∅, Y, {a}, {b}}. Therefore ℓc(τ)_{Y} = {∅, Y, {b}} and ℓc(τ_{Y}) = {∅, Y, {a}, {b}}. Hence ℓc(τ_{Y}) ≠ ℓc(τ)_{Y.}

3. Some continuity results

Now we give some definitions. Let P denote a property, not necessarily topological, possessed by certain subsets of a topological space.

Definition 27. [13] Let X be a topological space and A ⊆ X. Then

(1) A is a P -set if A possesses property P; and

(2) A has P -complement if X − A possesses property P.

Definition 28. Let f: X → Y be a function from a topological space X into a topological space Y. Then f is said to be

(1) P -continuous [12] if for each x ∈ X and each open set V containing f(x) and having P -complement, there is an open set U containing x such that f(U) ⊆ V; and

(2) P^{∗}-continuous [14] if for each x ∈ X and set B with f(x) ∈ intB, whenever B has P -complement, there is an open set U containing x such that f(U) ⊆ B.

The class of P^{∗}-continuous functions constitutes a subclass of the class of P - continuous functions, and they coincide with each other if the property P implies the property of being a closed set.

In the above definition if P denotes the property of being a closed LC-subspace, then we have the following definition.

Definition 29. A function f: X → Y is called ℓc-continuous if for each point x ∈ X and each open set V containing f(x) and having LC-complement there is an open set U containing x such that f(U) ⊆ V.

It is immediate from these definitions that every continuous function is ℓccontinuous. However, the reverse implication does not hold in general.

For example, let X = Y = {a, b, c, d}, ϑ = {Y, ∅, {b, c, d}}, τ = {X, ∅, {a}, {b, c, d}}. Let f be the identity mapping from (X, ϑ) onto (Y, τ). Then f is ℓccontinuous but it is not continuous.

Theorem 30. Let f: X → Y be an ℓc-continuous function such that f(X) is contained in a closed LC-subspace of Y. Then f is continuous.

Proof. Let L be a closed LC-subspace of Y containing f(X) and V be any open subset of Y. Then V ∪(Y −L) is an open subset of Y having LC-complement. Now the result follows since f is ℓc-continuous and f^{−1}(V) = f^{−1}(V ∪ (Y − L)).

The proof of the following result is straightforward, so we omit it.

Theorem 31. Let f: X → Y be a function from a topological space X into a topological space Y. The following statements are equivalent.

(1) f is ℓc-continuous,

(2) for every A of Y having LC-complement, f^{−1}(intA) ⊆ intf^{−1}(A).

There are more results that are related to ℓc-continuous functions that are similar to section 3 of [14].

The following Theorem is the ℓc-continuity version of a result given for ℓcontinuous functions in [7]. Its proof follows from the definitions.

Theorem 32. The function f : (X, ϑ) → (Y, τ) is ℓc-continuous if and only if f : (X, ϑ) → (Y, ℓc(τ)) is continuous.

Now we give a stronger version of ℓc-continuity similar to strong ℓ-continuity in [7].

Definition 33. A function f : (X, ϑ) → (Y, τ) is called strongly ℓc-continuous if f : (X, ℓc(ϑ)) → (Y, ℓc(τ)) is continuous.

It is clear from Theorem 32 that strong ℓc-continuity implies ℓc-continuity. On the other hand, ℓc-continuity does not imply strong ℓc-continuity in general.

Example 34. Let X = {a, b, c, d, e}, ϑ = {X, ∅, {a, b, c}, {d, e}}. Let Y = {a, b, c, d}, τ = {Y, ∅, {a, b, c}, {d}}. Let f: (X, ϑ) → (Y, τ) be a function defined by f(a) = a, f(b) = b, f(c) = c, f(d) = d, f(e) = d. Then f is ℓc-continuous but it is not strongly ℓc-continuous.

The following result is immediate from Theorem 32 and Definition 33.

Proposition 35. If f : (X, ϑ) → (Y, τ) is ℓc-continuous and g : (Y, τ) → (Z, υ) is strongly ℓc-continuous then g ∘ f is ℓc-continuous.

Acknowledgement. The authors would like to thank Professor Ivan L. Reilly for his helpful discussions during preparation of this paper.

REFERENCES

[1] Dontchev, J. and Ganster, M., On the product of LC-spaces, Q&A in General Topology, 15(1) (1997), 71–74.

[2] Dontchev, J., Ganster, M. and Kanibir, A., On some generalizations of LC-spaces, Acta Math. Univ. Comenianae, 57,2 (1999), 345–353.

[3] Dontchev, J., Ganster, M. and Kanibir, A., On locally LC-spaces, Mem. Fac. Sci. Kochi. Univ.(Math.) 21 (2000), 73–77.

[4] Ganster, M., Kanibir, A. and Reilly, I.L., Two comments concerning certain topological spaces, Indian J. Pure Appl. Math., 29(9) (1998), 965–967.

[5] Gauld, D.B., c-continuous functions and cocompact topologies, Kyungpook Math. J., 18 (1978), 151–157.

[6] Gauld, D.B., Topologies related to notions of near continuity, Kyungpook Math. J., 21 (1981), 195–204.

[7] Gauld, D.B., Mršević M., Reilly I.L. and Vamanamurthy M.K., CoLindelöf topologies and ℓ-continuous functions, Glasnik Matematički, 19(39) (1984), 297–308.

[8] Hdeib, H.Z., A note on L-closed spaces, Q&A in General Topology, 6 (1988), 67–72.

[9] Hdeib, H.Z. and Pareek, C.M., On spaces in which Lindelöf sets are closed, Q&A in General Topology, 4 (1986), 3–13.

[10] Henrikson, M. and Woods, R.G., Weak P -spaces and L-closed spaces, Q&A in General Topolgy, 6 (1988), 201–207.

[11] Kohli, J.K., A class of mappings containing all continuous mappings, Glasnik Matematički, 16(36) (1981), 361–367.

[12] Kohli, J.K., A unified approach to continuous and certain non-continuous functions, Symposium on General Topology and its Applications, Delhi (1978).

[13] Kohli, J.K., A unified view of (complete) regularity and certain variants (complete) regularity, Canad. J. Math., 36 (1984),783–794.

[14] Kohli, J. K., A framework including the theories of continuities and certain non-continuous functions, Note di Math., 10, 1 (1990), 37–45.

[15] Levy, R., A non-P L-closed space, Q&A in General Topology, 4 (1986),145–146.

[16] Misra, A.K., A topological view of P -spaces, Topology & Appl., 2 (1972), 349–362.

[17] Mukherji, T.K. and Sarkar, M., On a class of almost discrete space, Mat. Vesnik, 3(16)(31) (1979), 459–474.

[18] Wilansky, A., Between T_{1} and T_{2}, Amer. Math. Monthly., 74 (1967), 261–266.

(received 02.02.2006, in revised form 07.06.2007)

Department of Mathematics, Hacettepe University, 06532 Beytepe, Ankara, Turkey

E-mail: kanibir@hacettepe.edu.tr