β*- Closed and β*Open Maps in Topological Spaces

*- Closed and *Open Maps in Topological Spaces

*- Closed And *Open Maps In Topological Spaces

1. Antony Rex Rodgio And Jessie Theodore

   Department of mathematics,V.O.Chidambaram College,Tuticorin-62800Tamil Nadu,INDIA Ph.D. Scholar ,Department of mathematics,Sarah Tucker college,Tirunelveli-627007,Tamil

   Nadu,INDIA .

   Abstract

   The aim of this paper is to introduce the notions of *-closed maps, *-open maps and semi-*closed maps .Their relationships with other closed maps are investigated..It is found that the concept of *closed maps-are stronger than the concept of gspclosed maps.However it is weaker than closed maps. It is shown that the composition of *closed maps need not be *closed .The applications of these maps in some topological spaces are also studied.Also ultra *regular space and ultra *-normal spaces are introduced.

   1. 1. oiri,H.Maki and J.Umehara [9] introduced the concept of gp- closed and pre gp-closed map using gp-closed sets.Lellis Thivagar [6] introduced the concept of quasi -open and strongly -open map mappings using -open sets.Here we have introduced the concept of *-closed maps and semi *closed maps using *closed sets.Their respective open maps are also introduced

   2. Throughout this paper spaces( X ,)and (Y,) mean topological spaces and f: XY represents a single valued map .The following definitions and Theorems are useful in the sequel.

      A subset A of a topological space (X, ) is called

      1. A semi-open set [7] if Acl(int(A)) and a semi-closed set if int(cl(A))A,

      2. An -open set[6] if Aint(cl(int(A))) and an -closed set if cl(int(cl(A)))A,

      3. A semipre open set [2] (= -open set [1 ]) if Acl(int(cl(A))) and a semi-pre closed set(= closed) if int(cl(int(A)))A and

      The intersection of all semi-closed subsets of (X, ) containing A is called the semi-closure of A and is denoted by scl (A). Also the intersection of all closed (resp. semi-pre closed ) subsets of (X, ) containing A is called the closure (resp. semi-pre closure ) of A and is denoted by cl(A) (resp. spcl(A) ). Definition-2.2

      A subset A of a topological space (X, ) is called

      1. A generalized closed set (briefly g-closed)[8] if cl(A) U whenever A U and U is open in (X, ).

      2. A generalized semi-pre closed set (briefly gsp-closed) [5] if spcl(A)

         U whenever A U and U is open in (X, ).

      3. An -closed set [10] if cl(A) U whenever A U and U is semi- open in (X, )

      4. An * closed set [3] if spcl(A)U whenever AU and U is -open in (X, )

      5. A *-closed set [4] if spcl(A) int(U) whenever AU and U is -open

      The complement of g-closed(resp-gsp-closed, -closed, *closed,*closed) set is said to be g open (resp. gsp-open, -open, *-open,*open).

      1. A T space [10] if every -closed subset of (X, ) is closed (X, )

         A map f: XY is said to be

         1. g closed [8] (resp. g-open) if f(V) is g-closed (resp. g-open) in (Y, ) for every closed (resp.open) set V (X, )

         2. -closed [10] (resp.-open) if f(V) is -closed in (Y, ) for every set V of (X, )

         3. gsp-closed (resp.gsp-open)[5] if f(V) is gsp-closed in (Y, ) for every closed set V of (X, )

         4. *-closed(resp. *-open) [3] if f(V) is *-closed in (Y, ) for every closed set V of (X, )

            Theorem 2.4[4]: (i) Every *closed set is gsp closed(resp. *-closed) set.

            1. *closed set is independent of gclosed(resp. -closed) set.

            2. In a topological space X if the set of all *-open sets is closed under any union then *cl(A) is a *-closed set for every subset A of X.

Definition -3.1.1: A map f: XY is said to be *-closed, if the image of every closed set of X is *-closed in Y.

Definition -3.1.2: A map f: XY is said to be semi * closed if the image of every semi closed set of X is *-closed in Y.

Remark 3.1.3 If f: XY is closed ( semi *-closed), then f is *-closed since every closed (resp,semi closed) sets are *-closed .However the converses are not true. The following examples prove them.

{b, c},Y}. The identity map f: (X, )(Y, ) is *-closed but not closed. Since

{c} is closed in X but f({c})= {c} is not closed in Y.

Example 3.1.5: Let X=Y={a, b, c}, ={, {a},{b}, {a, b},X} and ={,{a}, Y}. The identity map f: (X, )(Y, ) is *-closed but not semi *-closed. Since {a} is semi closed in X but f({a})= {a} is not * closed in Y.

Proposition 3.1.6: Every *-closed map f: (X, )(Y, ) is a gsp closed (resp * closed) map.

Proof: Since every *-closed set is a gsp closed set (resp. * closed set) the proof follows by Theorem 2.4(i).

However the converses are not true. It can be seen through the following example.

Example -3.1.7 : Let X={a, b, c} = Y, ={, {a}, {a, b},x} and ={, {a}, {b, c}, Y}.Let f: (X, )(Y, ) be the identity map. Then f is a * closed and a gsp closed map but f is not *-closed, since for the closed set {c}, f({c})={c} is not *-closed in Y.

Theorem 3.1.9: A surjective map f: XY is *-closed if and only if for each subset S of Y and each open set U containing f -1(S), there exists a *-open set V of Y such that SV and f -1(V) U.

Proof: Suppose f is *-closed. Let S be any subset of Y and U be an open set of X containing f -1(S). If we let V=(f(Uc))c then V is *open in Y containing S and f -1(V) U.Conversely let F be any closed set of X. Let B = (f(F))c,then we have f -1 (B) Fc and Fc is open in X. By hypothesis there exists a *-open set V of Y such that BV and f -1(V) Fc and so F (f -1(V))c = f -1(Vc). Therefore, we obtain f(F) = Vc. Since Vc is *closed, f(F) is *-closed in Y. This gives f is *- closed.

{a},{b}, {a, b },Y} and ={,{a}, {a, b}, z}. Let f: (X, )(Y, ) and g:(Y,)(Z,) be identity maps. Then f and g are *-closed maps but gof: XZ is not *-closed. Since {a} is closed in X but gof({a}) = g(f({a})) = g({a}) ={a} is not *-closed in Z.

Proposition 3.1.12: If f: XY and g: YZ are *-closed maps with Y as a T* space then gof: XZ is a *-closed map.

Here *-open maps in topological spaces have been introduced and also obtained the characterizations of these maps.

Definition 3.2.1: A map f: XY is said to be *-open map if the image f(A) is * open in Y for every open set A in X.

Theorem -3.2.2: Every *-open map is a gsp open (resp. *) map but not conversely.

Proof: Since every *-open sets is a gsp open set (resp. * open set) the proof follows.

Example-3.2.3:Let X=Y= {a,b,c}, ={,{a}, {a,b}, X} and ={, {a,b}, Y}. Let f:XY be the identity map. Then f is *-open.

Theorem 3.2.4: For any bijection f: X Y the following statements are equivalent.

1. f -1: YX is *-continuous

2. f is a *-open map and

3. f is a *-closed map.

Proof: 12: Let U be an open set of X. By assumption (f -1)-1(U) = f(U) is *- open in Y and so f is a *-open map.

23: Let V be a closed set of X. Then Vc is open in X. By assumption f(Vc) = (f(v))c is *-open in Y and therefore f(V) is *-closed in Y. Hence f is a *- closed map.

31: Let V be a closed set of X. By assumption f(V) is *-closed in Y. But f(V)= (f -1)-1(V) and therefore f -1 is * continuous.

Theorem 3.2.5: Let f: XY be a mapping. If f is a *-open mapping, then for each xX and for each neighbourhood Uof x in X, there exists a * neighbourhood W of f(x) in Y such that Wf(U).

Proof: Let xX and U be an arbitrary neighbourhood of x. Then there exists an open set V in X such that xVU. By assumption f(V) is a *-open set in Y. Further f(x)f(V)f(U). Clearly f(U) is a *-neighbourhood of f(x) in Y and so the theorem follows if we take W=f(V).

Theorem 3.2.6: A function f: XY is *-open if and only if for any subset B of Y and for any closed set S containing f -1(B), there exists a *-closed set A of Y containing B such that f -1(A) S.

Definition 3.3.1 A space X is said to be ultra *-regular if for each closed set, F of X and each point xF there exist disjoint *-open sets U and V such that FU and xV.

1. X is ultra *-regular.

2. For every point x of X and every open set V containing x, there exists a * – open set A such that xA *cl(A) V.

ab. Let xX and V be an open set containing x. Then Vc is closed and xVc By (a) there exist disjoint *-open sets A and B such that x A and Vc B. that is Bc V. Since every open set is *-open, V is *-open. Since B is *-open, Bc is *-closed. Therefore *cl(Bc)V. Also since AB = . ABc. Therefore xA *cl(A) *cl(Bc) V. Hence, xA*cl(A)V.

ba: Let F be a closed set and xF. This implies that Fc is an open set containing x. By (b) there exists a *-open set A such that xA*cl(A)Fc. That is F(*cl(A))c. By Theorem 2.4(iii) *cl(A) is *-closed. Hence (*cl(A))c is *-open. Therefore, A and (*cl(A))c are the required * – open sets.

Theorem 3.2.3: Assume that *o() is closed under any union. If f: XY is a continuous open *-closed surjective map and X is a regular space, then Y is ultra *-regular.

Proof: Let yY and V be an open set containing y of Y. Let x be a point of X such that y=f(x). Since f is continuous, f -1(V) is open in X. Since X is regular there exists an open set U such that xUcl(U)f -1(V). Hence y=f(x)f(U)f(cl(U))V. Since f is a *-closed map f(cl(U)) is a *-closed set contained in the open set V. Therefore *cl(f(cl(U))) = f(cl(U))V. This implies that y f(U)*cl(f(U))*cl(f(cl(U)))V. Since f is an open map and U is open in X, f(U) is open Y. Since every open set is *-open, f((U) is *-open in

1. Thus for every point y of Y and every open set V containing y, there exist a *-open set f(U) such that yf(U)*cl(f(U))V. Hence by theorem 5.2.16 Y is ultra *-regular.

   Definition-3.2.4: A space X is said to be ultra *-normal if for each pair of disjoint closed sets A and B of X there exist disjoint *-open sets U and V such that AU and BV.

   Theorem -3.2.5: Assume that *o() is closed under any union. If f: XY is a continuous *-closed surjection and X is a normal space, then Y is ultra *- normal.

   Proof: Let A and B be disjoint closed sets of Y. Since X is normal there exist disjoint open sets U and V of X such that f -1(A)U and f -1(B) V. By theorem 3.1.10, there exist *-open sets G and H such that AG, BH and f -1(G)U and f -1(H) V. Then we have f -1(G)f -1(H) = and hence GH = . Since G is *-open, A G implies A* int (G) Similarly B* int (H). Therefore * int (G)* int (H) = . Thus Y is ultra * normal.

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