APPLICATION ON LOCAL DISCRETE EXPANSION

The process of changing a topology by some types of its local discrete expansion preserves s-closeness, S-closeness, semi-compactness, semi-T, semi-R, E {0,1,2}, and extremely dis- connectness Via some other forms of such above replacements one can have topologies which satisfy separation axioms the original topology does not have


INTRODUCTION
Throughout the present paper (X, 7-) is a topological space (or simply a space X) on which no separation axioms are assumed unless explicitly stated.For any B C X, clTB (resp int7 B) denotes the closure (resp interior) of/3 A subset B is said to be regular open (resp regular closed) if B int, (clT(B)) (resp B -clT (int(B))) A subset B of a space X is said to be r-semi open [12] (resp 7-- regular semi-open [2]) if there exists a r-open (resp.r-regular open) set U satisfying U c/3 c clTU B is r-semi-closed [3] if the set X-B is r-semi-open.The family of all regular open (resp regular semi- open, semi-open) sets in X is denoted by RO(X, 7-) (resp RSO(X, 7-),SO(X, 7-)) The union (resp intersection) of all 7--semi-open (resp r-semi-closed) sets contained in B (resp containing/3) is called the 7--semi-interior [3] (resp 7--semi-closure [3]) of B, and it is denoted as s-intB (resp s cluB) A space X is said to be extremely disconnected (denoted by E.D if for every open set U of X, clU is open in 7- The concept of local discrete expansion of a topology was first introduced by S P Young in 1977 [17], "Let (X, 7-) be a topological space and A be any subset of X The topology 7-[A] {U-H U E 7-, H c A} is called the local discrete expansion of 7-by A A space X is semi- T2 [13] (resp semi-T [1]) iff for x, y E X, x :/: y there exist U and V SO(X, 7-),x U and y V such that U n V (resp cl.U fqclV---).Semi-T0 and semi-T1 were introduced to topological spaces [13] by replacing the word "open" by "semi-open" in the definitions ofT0 and T1 respectively A space X is semi-R0 [6] iff for each semi-open set U and x E U, s cl {x} c U A space X is semi-R1 [6] iff for x, y E X such that s-clT{x} : s-cl{y} there exist disjoint semi-open sets U and V such that s-clT{x} c U, and s-cl.{y}C V. A space X is called cid [15] if every countable infinite subspace of X is discrete.A space X is semi-compact [7] (resp s-closed [5], S-closed [16]) if for every cover {V,:i I} of X by semi-open sets of X, there exists a finite subset I0 of I such that X t2 {V E I0} (resp X t2 scl(V,): E Io},X t2 cl(V): E I0}).
REMARK 1.1.For a subset A of a space (X, 7-) we say that A satisfies condition (C1) if A t_J U , for every U 7-{X}.
Listed below are theorems that will be utilized in this paper THEOREM 1.1 [14] If 7-and 7-' are two topologies on X such that 7-c 7-', then RO(X, T) RO(X, 7-') iffclG cl,,G for every G 7-' [equivalent iffint, F int-,F, for every F THEOREM 1.2 [11] If X is a space, and A c X satisfying (C1) Then, climlG cloG, for every G E 7-[A] THEOREM 1.3 [4] If X is a space, and A E SO(X,7-) such that A C B c cl,A Then.
B SO(X, 7-)  THEOREM 1.4 [10] IfX isa space, and B C X, then s cl, B BUint,cl, B THEOREM 1.5 [8] A space X is E D iff for every pair U and V of disjoint 7--open sets, we have clU cl, V THEOREM 1.6 [5] A space X is s-closed iffevery cover of X by regular semi-open sets has a finite subcover THEOREM 1. 7 15] (a) A space X is cid if every countable infinite subset is closed (b) Any infinite cid space is T THEOREM 1.$ 17] Let A be any subset of X Then (A, 7. THEOREM 1.10 [9] Let X be a T-space Then X is cid iff countable subsets have no limits points 2. ON LOCAL DISCRETE EXPANSION THEOREM 2.1.If (X, "r) is a space and A c X, then (i) SO(X, 7-[A]) C {t3-H: 13 80(X,7.),HC A} (ii) If A satisfying (C), then the inclusion symbol in (i) is replaced by equality sign PROOF.(i) Let W SO(X,-r[A]), then there exists V 7.
PROOF.In general SO(X, 7-) c SO(X, T[A]).To prove the converse, let W SO(X, T[A]), There are two cases.
(a) U X, then U H U Since cltAlU cl, U, then W SO(X, 7-).
Since AfqU=, then cl-A c (X U), and cl, A O U , implies to cl, H fq U , for each U e T {X} Hence U cl,H, and int.cl,H , and H is a T-semi-closed set Thus (X-H) SO(X,T) From Theorem 3, W SO(X, T) (ii) By Theorems 1.1 and 2, the proof is obvious COROLLARY 2.1.If X is a space, and A c X satisfying (C1) Then PROOF.By Theorem (2 2), the proof is obvious TIIEOREM 2.3.If X is a space, and A c X satisfying (C1).Then s cl,-[A]G s cl.,-G, for every G T cl-G [by Theorems 1, 12 and 14] THEOREM 2.4.If X is a space, and A c X satisfying (C1).Then (X, 7.) is E.D. iff (X, 7. [A]) is E D PROOF.Let (X, 7.) be E.D., W E 7.
[A] Then W U H, U E -, H c A. But cl.
The excluded point topology on an infinite set X is the family consisting of and all subsets of X not containing a point p of X. EXAMPLE 3.5.The excluded point topology is L-T1 and not L-T2 (also is an example of Q L T1 but not T1).
PROOF.If X is an infinite set and p is the excluded point and A C X, then: (i) Ifp A, we have 7-[A] T U {X B B C A}. Thus 7-[A] is T1 but not T2.[A]-7-L{X-B-Bc A} EXAMPLE 3.6.Let X-[0,1] and 7.={q,X, AcX.X-A is finite} If we takeS= (0,1], then "r[S] is the Discrete space This example is Q L T_, but not T.,, THEOREM 3.1.(X, 7.) is cid space ifft.7.
[A] whenever A is a countable infinite subset of X PROOF.We assume that (X, 7.) is cid, then A is closed and discrete subspace By Theorem 9 we have that 7-"r[A] Conversely we assume that "r "r[A] By Theorem 8, we have that (A, "r FI A) is a discrete subspace of X and (X, 7-) is cid space TtIEOREM 3.2.Every space (X, 7.) is L To PROOF.Assume that zo X We aim to prove that 7-IX-{To}] is To For this purpose let z,t X,x /, ifU G "r is an open set containing z, then U-{/} is an open set in "r[X-{zo}] and not containing If x0 :r, then X-{//} is an open in 7.[X-{x0}] and not containing V This completes the proof The following example illustrates a Q L space but not T2 EXAMPLE 3.7.(Countable complement topology [16]) If X is an uncountable set, we define the topology of countable complements on X by declaring open all sets whose complements are countable, together with $ and X (X, 7-) is T but not T2 Let A C X such that X-A is countable For x0 X A, A tO {To} is T-open, and so (A tO {:r0}) A {To} E r[A] For a:o A, A is r-open, which means that A (A {To}) {a:0} is r[A]-open Thus r[A] is discrete and consequently T2 UNSOLVED PROBLEM.If (X, "r) is a space which does not have a property P, what are the properties of the subset A that make (X, r[A]) have P (for P fixed property)

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ii) Ifp E A, then A is closed, and there are two cases (a) If B c A, p E B in this case any open set in -[A] is open in 7., e 7. -[A] (b) IfBc A,pBas(i) Thus7.