STRONG FORM OF PRE-I-CONTINUOUS FUNCTIONS

ed in Mathematical Reviews, Zentralblatt MATH &Tübitak Ulakbim Faculty of Sciences, Ankara University 06100 Beşevler, Ankara – Turkey ISSN 1303-5991


Introduction
In 1990, Jankovic and Hamlett [14] have de…ned the concept of I-open set via local function which was given by Vaidyanathaswamy [25].The latter concept was also established utilizing the concept of an ideal whose topic in general topological spaces was treated in the classical text by Kuratowski [16].In 1992, Abd El-Monsef et al [1] studied a number of properties of I-open sets as well as I-closed sets and Icontinuous functions and investigated several of their properties.In 1999, Dontchev [10] has introduced the notion of pre-I-open sets which are weaker than that of Iopen sets.In this paper, a new class of functions called strongly pre-I-continuous functions in ideal topological spaces is introduced and some characterizations and several basic properties are obtained.

Preliminaries
Throughtuout this paper, for a subset A of a topological space (X; ), the closure of A and interior of A are denoted by (A) and (A), respectively.An ideal topological space is a topological space (X; ) with an ideal I on X, and is denoted by (X; ; I), where the ideal is de…ned as a nonempty collection of subsets of X satisfying the following two conditions.(i) If A 2 I and B A, then B 2 I; (ii) If A 2 I and B 2 I, then A [ B 2 I.For a subset A X, A (I) = fx 2 Xj U \ A = 2 I for each neighbourhood U of xg is called the local function of A with respect to I and [14].When there is no chance of confusion, A (I) is denoted by A .Note that often X is a proper subset of X.For every ideal topological space (X; ; I), there exists topology (I), …ner than , generated by the base (I; ) = fU nI j U 2 and I 2 Ig, but in general (I; ) is not always a topology [14].Observe additionally that (A) = A [ A de…nes a Kuratowski closure operator for (I).A subset S of an ideal topological space (X; ; I) is said to be pre-I-open [10] (resp.semi I-open [12], *-dense-in-itself [13]) if S ( (S)) (resp.S ((S)), S S ).The complement of a pre-I-open set is called pre-I-closed [10].The intersection of all pre-I-closed sets containing S is called the pre-I-closure [26] of S and is denoted by P I (S).A set S is pre-I-closed if and only if P I (S) = S.The pre-I-interior [26] of S is de…ned by the union of all pre-I-open sets of (X; ; I) contained in S and is denoted by P I (S).The family of all pre-I-open (resp.pre-I-closed, semi-I-open) sets of (X; ; I) is denoted by P IO(X) [26] (resp.P IC(X); SIO(X)).The family of all pre-I-open (resp.pre-I-closed) sets of (X; ; I) containing a point x 2 X is denoted by P IO(X; x) (resp.P IC(X; x)).De…nition 2.1.A subset A of a topological space (X; ) is said to be: (i) semiopen if A ((A)) [17].

Strongly Pre-I-continuous functions
De…nition 3.1.A function f : (X; ; I) !(Y; ) is said to be strongly pre-I- It is clear that every strongly pre-I-continuous function is pre-I-continuous.But the converse is not always true as shown in the following example.
Theorem 3.4.Let (X; ; I) be -space.Then the function f : (X; ; I) !(Y; ) is strongly pre-I-continuous if and only if it is strongly precontinuous.
Recall that a topological space (X; ) is said to be submaximal if every dense subset of X is open.
Theorem 3.6.Let f : (X; ; I) !(Y; ) be a function.Then (i) If I = f?g, then f is strongly pre-I-continuous if and only if it is strongly precontinuous; (ii) If I = P (X), then f is strongly pre-I-continuous if and only if it is strongly semicontinuous; (iii) If I = N (= nowhere dense subsets of (X; )), then f is strongly pre-Icontinuous if and only if it is strongly precontinuous; (iv) If (X; ) is submaximal and I is any ideal on X, then f is strongly pre-Icontinuous if and only if it is strongly semicontinuous.
Theorem 3.7.For a function f : X !Y , the following are equivalent: Thus, f is strongly pre-I-continuous.Lemma 3.8.[23] Let (X i ; i ) i2^b e any family of topological spaces.Let X = i2^Xi , let A in be any subset of X n , n 2 ^, for each n = 1 to m.Let A = m n=1 A in 6 =in X be any subset of X.Then is semiopen set in X if and only if A in is semiopen set in X in , for each n = 1 to m. Theorem 3.9.A function f : (X; ; I) !(Y; ) is strongly pre-I-continuous, if the graph function g : (X; ; I) !X Y , de…ned by g(x) = (x; f (x)) for each x 2 X, strongly pre-I-continuous.
Y by Lemma 3.8 and contains g(x).Since g is strongly pre-I-continuous, there exists a pre-I-open set U of X containing x such that g(U ) X V .This shows that f (U ) V .By Theorem 3.7, f is strongly pre-I-continuous.
Proof.Let A i be an arbitrary semiopen set of Y i .Since P i is continuous and open, it is irresolute [[8], Theorem 1.2] and hence Hence, P i f is strongly pre-I-continuous for each i 2 ^.
Recall that a subset A of X is said to be -perfect if A = A [13].A subset of X is said to be I-locally closed if it is the intersection of an open subset and a -perfect subset of X [9].An ideal space (X; ; I) is I-submaximal if every subset of X is I-locally closed [4].! (Y; ) is said to be strongly irresolute if f 1 (V ) is semi-I-open in (X; ; I) for every semiopen set V of Y .De…nition 3.12.An ideal space (X; ; I) is said to be P -I-disconnected [4] if the ?6 = A 2 for each A 2 .Proposition 2. If f : (X; ; I) !(Y; ) is a strongly irresolute function and (X; ; I) is a P -I-disconnected space, then f is strongly pre-I-continuous.
Proof.Let V be any semiopen set of (Y; ).Since f is strongly pre-I-continuous, we have f 1 (V ) is pre-I-open in (X; ; I).Since A is semiopen in (X; ), by Proposition 2.10(V) of [9], (f jA ) 1 (V ) = A \ f 1 (V ) is preopen in A and hence f jA is strongly precontinuous.
Recall that a function f : (X; ; I) !(Y; ) is said to be pre-I-irresolute if f 1 (V ) 2 P IO(X) for every preopen set V of Y [10].De…nition 3.14.An ideal space (X; ; I) is said to be pre-I-connected if X is not the union of two disjoint non-empty pre-I-open sets of X. De…nition 3.15.[24] A topological space (X; ) is said to be semiconnected if X cannot be expressed as the union of two nonempty disjoint semiopen sets of X. Theorem 3.16.For the functions f : (X; ; I) !(Y; ; J) and g : (Y; ; J) !(Z; ; K), the following properties hold: (i) If f is pre-I-continuous and g is strongly semicontinuous, then g f is strongly pre-I-continuous; (ii) If f is strongly pre-I-continuous and g is semicontinuous, then g f is pre-I-continuous; (iii) If f is strongly pre-I-continuous and g is irresolute, then g f is strongly pre-I-continuous; (iv) If f is pre-I-irresolute and g is strongly pre-I-continuous, then g f is strongly pre-I-continuous.
Proof.Follows from their respective de…nitions.
Proof.Suppose Y is not semi-connected.Then there exist non-empty disjoint semiopen subsets U and This shows that X is not pre-I-connected.This is a contradiction and hence Y is semi-connected.Proof.Suppose that (Y; ) is semi-T 1 .For any distinct points x and y in X, there exist V; W 2 SO(Y ) such that f (x) 2 V , f (y) = 2 V , f (x) = 2 W and f (y) 2 W . Since f is strongly pre-I-continuous, f 1 (V ) and f 1 (W ) are pre-I-open subsets of (X; ; I) such that x 2 f 1 (V ), y = 2 f 1 (V ), x = 2 f 1 (W ) and y 2 f 1 (W ).This shows that (X; ; I) is pre-I-T 1 .Theorem 3.24.If f : (X; ; I) !(Y; ) is a strongly pre-I-continuous injection and Y is semi-T 2 , then (X; ; I) is pre-I-T 2 .
Proof.For any pair of distinct points x and y in X, there exist disjoint semiopen sets U and V in Y such that f (x) 2 U and f (y) 2 V .Since f is strongly pre-Icontinuous, f 1 (U ) and f 1 (V ) are pre-I-open sets in (X; ; I) containing x and y, respectively.Therefore, f 1 (U ) \ f 1 (V ) = ?because U \ V = ?.This shows that the space (X; ; I) is pre-I-T 2 .
Proof.If x 2 E c , then it follows that f (x) 6 = g(x).Since (Y; ) is semi-T 2 , there exist V , W 2 SO(Y ) such that f (x) 2 V and g(x) 2 W and V \ W = ?. Since f is strongly semi continuous and g is strongly pre I-continuous, f 1 (V ) is open and g 1 (W ) is pre-I-open in X with x 2 f 1 (V ) and x 2 g 1 (W ).Put A x = f 1 (V ) \ g 1 (W ).By Theorem 2.1 of [9](ii), A x is pre-I-open.If a point z 2 A x , then f (z) 2 V and g(z) 2 W . Hence f (z) 6 = g(z).This shows that A x E c and hence E is pre-I-closed in (X; ; I).De…nition 3.26.A space (X; ) is said to be: (i) s-regular if each pair of a point and a closed set not containing the point can be separated by disjoint semiopen sets [19].(iii) semi-normal if every pair of disjoint closed sets of X can be separated by semiopen sets [18].Proof.Follows from their respective de…nitions.

Lemma 3 . 18 . [ 22 ]De…nition 3 . 19 .Theorem 3 . 20 .Theorem 3 . 23 .
For any function f : (X; ; I) !(Y; ), f (I) is an ideal on Y .Now, we recall the following de…nitions.An ideal space (X; ; I) is said to be pre-I-compact (resp.pre-I-Lindelöf, SI-compact[2], SI-Lindelof[2]) if for every pre-I-open (resp.pre-Iopen, semiopen, semiopen) cover fW : 2 4g on X, there exists a …nite (resp.countable) subset 4 0 of 4 such that X S fW : 2 4 0 g 2 I.If f : (X; ; I) !(Y; ; J) is strongly pre I-continuous surjection and (X; ; I) is pre I-compact, then Y is S-f (I)-compact.Proof.Let fV : 2 Og be a semiopen cover of Y , then ff 1 (V ) : 2 Og is a pre-I-open cover of X from strongly pre-I-continuity.By hypothesis, there exists a …nite subcollection, ff1 (V i ): i = 1, 2, .... ng such that X -S ff 1 (V i ): i = 1, 2,.... ng 2 I, implies , Y -S fV i : i = 1, 2, .... N g 2 f (I).Therefore, (Y; ) is S-f (I)-compact.Theorem 3.21.Let f : (X; ; I) !(Y; ) be a strongly pre-I-continuous surjection.If (X; ; I) is pre-I-Lindelöf, then (Y; ) is semi-f Proof.Similar to the proof of Theorem 3.20.De…nition 3.22.An ideal space (X; ; I) is said to be:(i) pre-I-T 1 if for each pair of distinct points x and y of X, there exist pre-Iopen sets U and V of (X; ; I) such that x 2 U and y = 2 U , and y 2 V and x = 2 V .(ii) pre-I-T 2 if for each pair of distinct points x and y in X, there exists disjoint pre-I-open sets U and V in X such that x 2 U and y 2 V .(iii) semi-T 1 if for each pair of distinct points x and y of X, there exist semiopen sets U and V of (X; ; I) such that x 2 U and y = 2 U , and y 2 V and x = 2 V [20].(iv) semi-T 2 if for each pair of distinct points x and y in X, there exist disjoint semiopen sets U and V in X such that x 2 U and y 2 V[20].If f : (X; ; I) !(Y; ) is a strongly pre-I-continuous injection and (Y; ) is semi-T 1 , then (X; ; I) is pre-I-T 1 .

De…nition 3 . 27 .
An ideal space (X; ; I) is said to be: (i) pre-I-regular if each pair of a point and a closed set not containing the point can be separated by disjoint pre-I-open sets.(ii) pre-I-normal if every pair of disjoint closed sets of X can be separated by pre-I-open sets.Theorem 3.28.Let f : (X; ; I) !(Y; ) be a strongly pre-I-continuous injection.Then the following properties hold: (a) If (Y; ) is semi-T 2 , then (X; ; I) is pre-I-T 2 , (b) If (Y; ) is semi regular and f is open or closed, then (X; ; I) is pre-Iregular, (c) If (Y; ) is semi normal and f is closed, then (X; ; I) is pre-I-normal.