 Open Access
 Total Downloads : 607
 Authors : J. Antony Rex Rodgio And Jessie Theodore
 Paper ID : IJERTV2IS70585
 Volume & Issue : Volume 02, Issue 07 (July 2013)
 Published (First Online): 29072013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
* Closed and *Open Maps in Topological Spaces
* Closed And *Open Maps In Topological Spaces

Antony Rex Rodgio And Jessie Theodore
Department of mathematics,V.O.Chidambaram College,Tuticorin62800Tamil Nadu,INDIA Ph.D. Scholar ,Department of mathematics,Sarah Tucker college,Tirunelveli627007,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 mapsare 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.


oiri,H.Maki and J.Umehara [9] introduced the concept of gp closed and pre gpclosed map using gpclosed 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


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

A semiopen set [7] if Acl(int(A)) and a semiclosed set if int(cl(A))A,

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

A semipre open set [2] (= open set [1 ]) if Acl(int(cl(A))) and a semipre closed set(= closed) if int(cl(int(A)))A and
The intersection of all semiclosed subsets of (X, ) containing A is called the semiclosure of A and is denoted by scl (A). Also the intersection of all closed (resp. semipre closed ) subsets of (X, ) containing A is called the closure (resp. semipre closure ) of A and is denoted by cl(A) (resp. spcl(A) ). Definition2.2
A subset A of a topological space (X, ) is called

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

A generalized semipre closed set (briefly gspclosed) [5] if spcl(A)
U whenever A U and U is open in (X, ).

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

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

A *closed set [4] if spcl(A) int(U) whenever AU and U is open
The complement of gclosed(respgspclosed, closed, *closed,*closed) set is said to be g open (resp. gspopen, open, *open,*open).

A T space [10] if every closed subset of (X, ) is closed (X, )
A map f: XY is said to be

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

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

gspclosed (resp.gspopen)[5] if f(V) is gspclosed in (Y, ) for every closed set V of (X, )

*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.

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

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.
Example3.2.3: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.

f 1: YX is *continuous

f is a *open map and

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.

X is ultra *regular.

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

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.
Definition3.2.4: 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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