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On Theta paracompactness by Grills.

[section] 1. Introduction

Choquet [3] introduced the concept of grills in 1947. The idea of grills was found to be very useful device like nets and filters. Also for the investigations of many topological notions like compactifications, proximity spaces, theory of grill topology was used.

The notion of paracompactness in ideals was initiated by Hamlett et al [6] in the year 1997. B. Roy and M. N. Mukherjee extended the concept of para compactness in terms of grills. Following their work we formulate the new definition of [theta]-paracompactness via grills. Also we attempted to achieve a general form of the well known Michaels theorem on regular paracompact spaces perticularly for [theta]-open sets.

[section] 2. Preliminaries

Definition 2.1. [3] A colletion G of nonempty subsets of a set X is called a grill if

(i) A [member of] G and A [subset or equal to] B [subset or equal to] X implies that B [subset or equal to] G, and

(ii) A [union] B [member of] G (A, B [subset or equal to] X) implies that A [member of] G or B [member of] G.

Definition 2.2. [8] Let (X, [tau]) be a topological space and G be a grill on X. We define a mapping [PHI] : P(X) [right arrow] P(X), denoted by [[PHI].sub.G] (A, [tau]) or simply [PHI](A), is called the operator associated with the grill G and the topology [tau], and is defined by [PHI](A)= {x [member of] X: A [intersection] U [member of] G, [for all] U [member of] [tau](x)}.

Definition 2.3. [8] The topology [tau] of a topological space (X, [tau]) is said to be suitable for a grill G on X if for any A [subset or equal to] X, A\[PHI](A) [??] G

Definition 2.4. [8] A grill G is called a [micro] grill if any arbitrary family {[A.sub.[alpha]]: [alpha] [member of] [LAMBDA]} of subsets of X, [[union].sub.[alpha]][A.sub.[alpha]] [member of] G then [A.sub.[alpha]] [member of] G for atleast one [alpha] [member of] [LAMBDA].

Definition 2.5. [7] A topological space (X, [tau]) is said to be G [theta]-regular if for any [theta]-closed set F in X with x [??] F, there exist disjoint [theta]-open sets U and V such that x [member of] U, and F\V [??] G.

Definition 2.6.[15] A paracompact space (X, [tau]) is a Hausdorff space with the property that every open cover of X has an open locally finite refiniment.

Definition 2.7.[151 In a Lindelof space (X, [tau]), for every open cover there exists a subcover of X, which is having countable collection of open sets.

[section] 3. G-paracompactness through t9-open sets

Definition 3.1. Let G be a grill on a topological space (X, [tau]). Then the space X is said to be [theta]-paracompact with respect to the grill or simply G-[theta]-paracompact if every [theta]-open cover U={[U.sub.[alpha]]: [alpha] [member of] [lambda]} of X has a precise locally finite [theta]-open refiniment U* such that X\[union] U* [??] G. Also a cover has a precise refinement means, there exists a collection V = {[V.sub.[alpha]] : [alpha] [member of] [LAMBDA]} of subsets of X such that V[alpha] [subset or equal to] [U.sub.[alpha]], for all [alpha] [member of] A.

Remark 3.1. (i) Every [theta]-paracompact space X is G-[theta]-paracompact, for every grill G on X.

(ii) For the grill G = P(X)\[empty set], the concepts of [theta]-paracompactness and G-[theta]-paracompactness coincide for any space X, where P(X) denotes the power set of X.

(iii) If [G.sub.1] and [G.sub.2] are two grills on a space with [G.sub.1] C [G.sub.2], then [G.sub.2]-[theta]- paracompactness of X ft [G.sub.1]-[theta]-paracompactness of X. Moreover, it may so happen that a space X is [G.sub.1]-[theta]- paracompact as well as [G.sub.2]-[theta]-paracompact while the grills [G.sub.1] and [G.sub.2] are non-comparable.

(iv) Considering G [theta] paracompactness, refinement need not be a cover.

Theorem 3.1. Let G be a [mu] grill on a topological space (X, [tau]). Then (X, [tau]G) is G-6paracompact if (X, [tau]) is so.

Proof. Let us consider a cover W of X by basic [theta]-open sets of (X, [tau]G), given by W = {[W.sub.[alpha]]: [alpha] [member of] [LAMBDA]}, where for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [A.sub.[alpha]] [member of] [tau] also [theta]-open set and [A.sub.[alpha]] [??] G. Then U = {[U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} is a [tau] [theta]-open cover of X. By G-[theta]- paracompactness of (X, [tau]), U has a [tau] locally finite [tau] [theta]-open precise refinement V = {[V.sub.[alpha]]: [alpha] [member of] [LAMBDA]} such that X\([[union].sub.[alpha][member of][LAMBDA]][V.sub.[alpha]])[??] G. It suffices to show that W* = {[V.sub.[alpha]]\[A.sub.[alpha]]: [alpha] [member of] [LAMBDA]} is a precise [[tau].sub.G]-locally finite [tau]G [theta] open refinement of W.

W* is a [[tau].sub.G] [theta]-open precise refinement of W. Also W is [tau]-locally finite and [tau] [subset not equal to] [tau]G, V is [tau]G-locally finite, and hence W* is [tau]G-locally finite. It thus remains to show that X\[[union].sub.[alpha][member of][LAMBDA]] ([V.sub.[alpha]]\[A.sub.[alpha]])[??] G. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The union over A stands for all possible partition (1) holds. Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (since [A.sub.[alpha]] [member of] G for each a). Furthermore, for any partition {[[LAMBDA].sub.1], [[LAMBDA].sub.2]} of [LAMBDA] with the property (1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence the result.

Theorem 3.2. Let G be a grill on a space (X, [tau]) such that [tau]\{[empty set]}[subset or equal to] G. If t is suitable for G and (X, [tau]G) is G-[theta]-paracompact, then (X, [tau]) is G-[theta]-paracompact.

Proof. Let U={[U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} be a [tau] [theta]-open cover of X. Then U is a [tau]G [theta]-open cover of X. Hence U is a [tau]G-locally finite precise refinement {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

We now show that V = {[V.sub.[alpha]]: [alpha] [member of] [LAMBDA]} is t locally finite. In fact, for each x [member of] X there exists some U [member of] [tau]G such that U[intersection]([V.sub.[alpha]]\[A.sub.[alpha]])=[empty set], for all [alpha] [not equal to] [[alpha].sub.1], [[alpha].sub.2], ... , [[alpha].sub.n](assumption). But U = V\A, where V [member of] [tau] and A [??] G. Thus for any [alpha] [not equal to] [[alpha].sub.1], [[alpha].sub.2], ... , [[alpha].sub.n], (V\A) n (Va\Aa) = 0, That is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then either [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We claim that V [intersection] [V.sub.[alpha]] = [empty set]. For otherwise, V [intersection] [V.sub.[alpha]] is nonempty [tau][theta]-open set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a contradiction. Thus V is [tau] locally finite.

Again, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence by (2), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Now W = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [tau] locally finite [tau][theta]-open precise refiniment of U such that X\(UW)[??] G. Thus (X, [tau]) is G-[theta]-paracompact.

Corollary 3.1. Let (X, [tau]) be a topological space and G a [mu]-grill on X such that [tau]\{[empty set]}[subset not equal to] G and [tau] is suitable for G. Then (X, [tau]) is G [theta]-compact iff (X, [tau]G) is G-[theta]-paracompact.

Corollary 3.2. [11] For any topological space X, Gs = {A [subset to equal to] X : intcl A [not equal to] 0} is a grill on X.

A weaker form of paracompactness is almost paracompactness and the definition for almost [theta]-paracompactness is,

Definition 3.2. A topological space (X, t) is said to be almost [theta]-paracompact if every [theta]-open cover U of X has a locally finite [theta]-open refinement U* such that X\cl([union] U*)= [empty set].

Theorem 3.3. A topological space (X, [tau]) is almost 6-para compact iff X is G[delta] [theta]-paracompact.

Proof. Let U be an [theta]-open cover of an almost [theta]-paracompact space (X, [tau]). Then there exists a precise locally finite [theta]-open refinement U* of U such that X\cl(U*)= [empty set]. We claim that X\([union] U*)[member of] G. For otherwise, X\([union] U*)[member of] G ft intcl(X\([union] U*))[not equal to] [empty set] [right arrow] X\clint([union] U*)[not equal to] [empty set] [right arrow] X\cl([union] U*)= [empty set], a contradiction. Thus (X, [tau]) is a G[delta] [theta]-paracompact.

We now prove a stronger converse that whenever G is any grill on X with [tau]\{[empty set]} [subset or equal to] G, then the almost [theta]-paracompactness of (X, [tau]) is implied by the G[theta]-paracompactness of X. We first observe that for such a grill G, we have intA = [empty set] whenever A([subset or equal to] X) [member of] G. Now let U be an [theta]-open cover of X. Then by the definition of G[theta]-paracompactness there exists a precise locally finite [theta]-open refinement U* of U such that X\([union] U*)[??] G. Thus int (X\([union] U*)) = [empty set], That is X = cl([union] U*), proving (X, [tau]) to be almost [theta]-paracompact.

[section] 4. Principal grill [A], its regularity and [theta]-paracompactness

Definition 4.1.[11] Let X be a nonempty set and ([empty set] [not equal to])A [subset or equal to] X. Then the principal [A] is defined as [A]= {B [subset or equal to] X : A [intersection] B [not equal to] [empty set]}.

Remark 4.1. In the grill topological space X, if G=[X], then [X]-[theta]-paracompactness reduces simply to [theta]-paracompactness.

Definition 4.2. G is a grill on a topological space (X, [tau]), the space X is said to be G-[theta] regular if for each [theta]-closed subset F of X and each x [member of] X\F, there exist disjoint [theta]-open sets U and V such that x [member of] U and F\V [??] G.

Remark 4.2. From the above two definitions the principal grill [X] generated by X is, in fact, P(X)\{[empty set]} and hence a space (X, [tau]) is [X]-[theta]-regular iff (X, [tau]) is [theta]-regular.

Remark 4.3. Every regular space is G-[theta]-reguler for any grill on X.

Theorem 4.1. Let X be any nonempty subset of a space (X, [tau]). Then (X, [tau]) is [A]-6regular iff for each [theta]-closed subset F of X and each x [??] F, there exist disjoint [theta]-open sets U and V such that x [??] U and F [intersection] A [subset or equal to] V.

Proof. Let (X, t) be a [A]-[theta]-regular and F a [theta]-closed subset of X and x [member of] X\F. Then there exist disjoint [theta]-open sets U and V such x [member of] U and F\V [??] [A]. Now, F\V [??] [A] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, let the given condition hold and let F be a [theta]-closed subset of X with x [subset or equal to] X\F. Then there exist disjoint [theta]-open sets U and V such that x [member of] U and F [intersection] A [subset not equal to] V. Now, F [intersection] A [subset or equal to] V [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We modify the E. Micheal's theorem for [theta]-open sets.

Theorem 4.2. Let G be a grill on a space (X, [tau]). If X is G-[theta]-paracompact and [sub.[theta]][[tau].sub.2] space, then X is G-[theta]-regular.

Proof. Let F be a 6-closed subset of X and y [member of] X\F. Then the Hausdorffnes of X implies that for each x [member of] F, there exist disjoint [theta]-open sets [G.sub.x] and [H.sub.x] such that y [member of] [G.sub.x] and x [member of] [H.sub.x]. Clearly y [member of] cl[H.sub.x]. Then U = {[H.sub.x]: x [member of] F}[union]{X\F} is an [theta]-open cover of X. Thus there exists a precise locally finite [theta]-open refinement U*= {[H.sub.x]: x [member of] F}[union]{W} such that [H.sub.x] [subset or equal to] [H.sub.x] for each x [member of] F, W [subset or equal to] X\F and X\([union] U*)[??] G. Let G = X\[union] {cl[H.sub.x]: x [member of] F}. Then G and H are two nonempty disjoint [theta]-open sets, such that y [member of] G, F\H [??] G. Hence (X, [tau]) is G-[theta]-regular.

Corollary 4.1. Let A be a nonempty subset of a space (X, [tau]). If X is an [A]-[theta]-paracompact Hausdorff space, then it is [A]-[theta]-regular.

Corollary 4.2. A [theta]-paracompact space is [theta]-regular. The proof is immediate.

Lemma 4.1. For a nonempty subset A of a Hausdorff space (X, [tau]), let X be [A]-[theta]-paracompact. Then for each x X and each [theta]-open set U containing x, there exists a [theta]-open neighbourhood V of x such that clV\U [subset or equal to] X\A. That is (clV\U)[intersection]A = [empty set], and hence clV[intersection]A [subset or equal to] U.

Proof. Let x [member of] X and U be an [theta]-open neighbourhood of x. Then X\U is a [theta]-closed subset of X, not containing x. As (X, [tau]) is [A]-[theta]-regular, by theorem 4.3, there exists two disjoint [theta]-open sets G and V such that x [member of] V and (X\U)[intersection]A [subset or equal to] G. Now, G [intersection] clV = [theta] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is clV\U [subset or equal to] X\A and hence the proof.

Theorem 4.3. Let (X, [tau]) be an [A]-[theta]-paracompact, Hausdorff space for some nonempty subset A of X and U= {[U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} be a [theta]-open cover of X. Then there exists a precise locally finite [theta]-open refinement {[G.sub.[alpha]]: [alpha] [member of] [LAMBDA]} of U such that A [subset or equal to] [union]{[G.sub.[alpha]]: [alpha] [member of] [LAMBDA]} and cl[G.sub.[alpha]][intersection]A [subset or equal to] [U.sub.[alpha]] [intersection] A.

Proof. Let U= {[U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} be a [theta]-open cover of X. Then by the Lemma 4.1, for each [alpha] [member of] [LAMBDA] and each x [member of] [U.sub.[alpha]], there exists [V.sub.[alpha],x] [member of] [tau] with x [member of] [V.sub.[alpha],x] such that [clV.sub.[alpha],x] [intersection] A [subset or equal to] [U.sub.[alpha]]. Now V={[V.sub.[alpha],x]: [alpha] [member of] [LAMBDA]} is a [theta]-open cover of X.

Hence by [A]-[theta]-paracompactness of X, there exists a precise locally finite [theta]-open refinement W={[W.sub.[alpha],x]: x [member of] [U.sub.[alpha]], [alpha] [member of] [LAMBDA]} of V such that X\([union] {[W.sub.[alpha],x] : x [member of] [U.sub.[alpha]], [alpha] [member of] [LAMBDA]})[??][A]. That is A [subset or equal to] [union]{[W.sub.[alpha],x]: x [member of] [LAMBDA]}. Now, for any x [member of] [U.sub.[alpha]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each [alpha] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each [alpha] [member of] [LAMBDA]. Then clearly {G[alpha]: [alpha] [member of] [LAMBDA]} is a precise locally finite [theta]-open refinement of U, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 4.4. Let (X, [tau]) be a Hausdorff space and A a dense subset of X. Then the following statements are equivalent:

(i) (X, [tau]) is [A]-[theta]-paracompact.

(ii) Each [theta]-open cover of X has a precise locally finite refinement that covers A and consists of sets which are not necessarily closed or open.

(iii) For each [theta]-open cover U={([U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} of X, there exists a locally finite 6-closed cover {[F.sub.[lambda]]: [alpha] [member of] [LAMBDA]} of X such that [F.sub.[alpha]] [intersection] A [subset or equal to] [U.sub.[alpha]] for each [alpha] [member of] [LAMBDA].

Proof. (i)ft(ii) It is trivial.

(ii) [right arrow] (iii) Let {[U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} be a [theta]-open cover of X. Then for any x [member of] X, there exists some [U.sub.[alpha](x)] [member of] U such that x [member of] [U.sub.[alpha](x)]. Then by Lemma 4.1, there exists some [H.sub.x] [member of] [tau] with x [member of] [H.sub.x] such that cl[H.sub.x] [intersection] A [subset or equal to] [U.sub.[alpha]](x). Thus H = {[H.sub.x]: x [member of] X} is a [theta]-open cover of X, and hence there is a precise locally finite refinement {[A.sub.x]: x [member of] X} of H such that A [subset not equal to] {[A.sub.x]: x [member of] X}. Since {[A.sub.x]: x [member of] X} is locally finite, so is {cl[A.sub.x]: x [member of] X}. Thus U{cl[A.sub.x]: x [member of] X}=[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [F.sub.[alpha]] is [theta]-closed for each [alpha] [member of] [LAMBDA], as it is a union of locally finite [theta]-closed sets. Thus {[F.sub.[alpha]]: [alpha] [member of] A} is locally finite and a cover of X. Finally [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for each [alpha] [member of] A.

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a [theta]-open cover of X. Let {[F.sub.[alpha]]: [alpha] [member of] [LAMBDA]} be a locally finite [theta]-closed cover of X such that [F.sub.[alpha]] n A [subset or equal to] [U.sub.[alpha]] for each a [member of] A. For any x [member of] X, there exists [V.sub.x] [member of] [tau] with x [member of] [V.sub.x] such that [V.sub.x] [intersection] [F.sub.[alpha]] [not equal to] [empty set] for atmost finitely many [alpha] [member of] [LAMBDA]. Now, V = {[V.sub.x]: x [member of] X} is a cover of X. So there exists a locally finite 6-closed cover {[B.sub.x]: x [member of] X} such that [B.sub.x] [intersection] A [subset or equal to] [V.sub.x], for all x [member of] X. Thus {[B.sub.x] [intersection] A : x [member of] A} is a cover of A.

Let us now consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We first note that U([F.sub.[alpha]]) is [theta]-open for each [alpha] [member of] [LAMBDA]. Now, [F.sub.[alpha]] [intersection] A [subset or equal to] U([F.sub.[alpha]]). In fact, y [member of] [F.sub.[alpha]] [intersection] A and y [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and y [member of] [B.sub.y'] for some y' [member of] X [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a contradiction.

We show that {U([F.sub.[alpha]]: [alpha] [member of] [LAMBDA])} is locally finite. Each x [member of] X has some [theta]-open neighbourhood W intersecting finitely many [B'.sub.x]s, say [B.sub.x1], [B.sub.x2], ... , [B.sub.xn]. Then W is contained in [U.sup.n.sub.i=1][B.sub.x4] (since {[B.sub.x]: [alpha] [member of] [LAMBDA]} is a [theta]-cover of X). Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Each [B.sub.x] [intersection] A is contained in [V.sub.x], where [V.sub.x] intersects atmost finitely many [F.sub.[alpha]] [right arrow] [B.sub.x] [intersection] A intersects atmost finitely many [F.sub.[alpha]] [right arrow] each set [B.sub.x] intersects atmost finitely many U([F.sub.[alpha]]) [right arrow] W intersects atmost finitely many U([F.sub.[alpha]]). Thus {U([F.sub.[alpha]]): [alpha] [member of] [LAMBDA]} is locally finite. Also {U([F.sub.[alpha]]): [alpha] [member of] [LAMBDA]} covers A, because [F.sub.[alpha]] [intersection] A [subset or equal to] [U.sub.[alpha]] and {[F.sub.[alpha]] [intersection] [U.sub.[alpha]]: [alpha] [member of] [LAMBDA]} is a [theta]-cover of A.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then U* is a precise locally finite [theta]-open refinement of U. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 4.3. In a regular space X, the following are equivalent:

(i) X is [theta]-paracompact.

(ii) Every [theta]-open cover of X has a locally finite refinement consisting of sets not necessarily [theta]-open or [theta]-closed.

(iii) Each [theta]-open cover of X has a closed locally finite refinement.

Theorem 4.5. Let G and G be two grills respectively on two topological spaces (X, t) and (Y, [tau]). Let f: (X, [tau]) [right arrow] (Y, [tau]) be a [theta]C homeomorphism and f (G) [contains not equal to] G'. If (X, [tau]) is G[theta]-paracompact then (Y, [tau]) is G'[theta]-paracompact. Here f(G) stands for {f(G): G [member of] G} which is clearly a grill in Y.

Proof. Let {[V.sub.[alpha]]: [alpha] [member of] [LAMBDA]} be a [theta]-open cover of Y. Then by continuity and surjectiveness of f, {[f.sup.-1]([V.sub.[alpha]]): [alpha] [member of] [LAMBDA]} is a [theta]-open cover of X. Hence by G[theta]-compactness of (X, [tau]), there exists a locally finite precise [theta]-open refinement {[W.sub.[alpha]]: [alpha] [member of] [LAMBDA]} of {[f.sup.- 1]([V.sub.[alpha]])} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since f is an [theta]C homeomorphism, f is bijective, f and [f.sup.-1] both are [theta]-irresolute maps. We have {f([W.sub.[alpha]]): [alpha] [member of] [LAMBDA]} is an [theta]-open precise refinement of {[V.sub.[alpha]]: [alpha] [member of] [LAMBDA]} in (Y, [tau]). We note {[W.sub.[alpha]]: [alpha] [member of] [LAMBDA]} is locally finite as f is a homeomorphism. Now, as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus (Y, [tau]) is G'[theta]-paracompact.

Corollary 4.4. Let (X, [tau]) and (Y, [tau]) be two topological spaces, A([not equal to] [empty set]) [subset not equal to] X, and f: (X, [tau]) [right arrow] (Y, [tau]) a homeomorphism. If (X, [tau]) is [A]-[theta]-paracompact then (Y, [theta]) is [f(A)][theta]-paracompact.

Proof. If we put [f(A)] = f([A]) then we can get the result using the previous theorem. Let A = X in the previous theorem we get the next Corollary.

Corollary 4.5. Let (X, [tau]) and (Y, [tau]) be two topological spaces. f: (X, [tau]) [right arrow] (Y, [tau]) a homeomorphism. If (X, [tau]) is [theta]-paracompact then (Y, [tau]') is [theta]-paracompact.

[section]5. Relations among other compactness with G-paracompact ness through [theta]-open sets

Definition 5.1. A space X is T-Lindelof if for every [theta]-open cover there exists a [theta]-open subcover for X, which is having countable collection of [theta]-open sets.

Theorem 5.1. Every G [theta]-regular, T-Lindelof space is G [theta]-paracompact.

Proof. Let (X, [tau]) be a [theta]-regular, T-Lindelof space. Let U be a [theta]-open cover of X. Since (X, [tau]) is a T-Lindelof, there exists a countable subcollection V of U that covers X. Then V is a [theta]-open refiniment of U. Since (X,t) is [theta]-regular the space (X,t) is G [theta]-paracompact.

Remark 5.1. (i) Every G [theta]-compact space is G [theta]-paracompact.

(ii) Every [sub.[theta]][T.sub.2], G [theta]-paracompact space is [theta]-normal.

Remark 5.2. From the above results we have the following implications:

[ILLUSTRATION OMITTED]

Note 5.1. From [5], [14], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and that [T.sub.[theta]] = [tau] if and only if (X, [tau]) is regular. So, if the space is regular the concept of [theta]-paracompactness coincide with paracompactness.

Acknowledgement

The authors are grateful to the referee for valuable suggestions and comments towards the imprevement of the paper.

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I. Arockia Rani ([dagger]) and A. Karthika ([double dagger])

Department of Mathematics, Nirmala College, Coimbatore
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Author:Rani, I. Arockia; Karthika, A.
Publication:Scientia Magna
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Date:Jun 1, 2012
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