A DECOMPOSITION OF CONTINUITY ON F*– SPACES AND MAPPINGS ON SA*– SPACES

SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 2008, 3(1), 51-59 A DECOMPOSITION OF CONTINUITY ON F*– SPACES AND MAPPINGS ON SA*– SPACES A.ACIKGOZ*, S. YUKSEL**, I. L. REILLY*** * Aksaray University Faculty of Arts And Sciences Department of Mathematics 68100 Aksaray, TURKEY ** Selcuk University Faculty of Arts And Sciences Department of Mathematics 42031 Campus – Konya / TURKEY e–mail: *** The University of Auckland Department of Mathematics New Zealand e–mail: Received: 28 September 2007, Accepted: 23 April 2008 Abstract: An ideal topological space (X,τ,I) is said to be an F* – space if A=Cl*(A) for every open set A ⊂ X. In this paper, a decomposition of continuity on F* – spaces is introduced. An ideal topological space (X,τ,I) is said to be an SA* – space if (A)*⊂ A for every set A⊂X. It is shown that δI – r – continuity (resp. pre – I – continuity, semi – δ – I – continuity, * – perfect continuity) is equivalent to R – I – continuity (resp. R – I – continuity, t – I – continuity, * – dense – in – itself continuity) if the domain is an SA* – space. Key words: R – I – open set, δ – I – open set, δ – I – regüler set, decomposition of R – I – continuity, topological ideal. Mathematics Subject Classification (2000): Primary 54C08, 54A20; Secondary 54A05, 54C10. F*-UZAYLARDA SÜREKLİLİĞİN BİR AYRIŞIMI VE SA* -UZAYLARDA DÖNÜŞÜMLER Özet: Eğer (X, τ, I) uzayının her açık A alt kümesi için A = Cl*(A) ise bu taktirde bu uzaya F* – uzay denir. Bu çalışmada, F* – uzayında sürekliliğin bir ayrışımı verildi. Eğer (X, τ, I) uzayının her açık A alt kümesi için (A)*⊂ A ise bu taktirde bu uzaya SA* –uzay denir. SA*-uzayında δI – r –süreklilik (sırasıyla, pre-I-süreklilik, semi-Isüreklilik, * – perfect süreklilik) ile R – I – sürekliliğin (sırasıyla, R – I – süreklilik, t – I – süreklilik, kendi içinde *-yoğun süreklilik) birbirine eşdeğer olduğu gösterildi. Anahtar Kelimeler: R-I-açık küme, δ – I – açık küme, δ – I – regüler küme, R – I – sürekliliğin ayrışımı, ideal topoloji. 1. INTRODUCTION Recently, ACIKGOZ et al. (2004) introduced the notion of a “δ − I − open set” in an ideal topological space, investigated some of its properties and obtained a decomposition of a α − I − continuous function using this set. HATIR & NOIRI A. ACIKGOZ, S. YUKSEL, I. L. REILLY 52 (2002) introduced the notions of t – I – sets, α* – I – sets, BI – sets and CI – sets. YÜKSEL et al. (2005) introduced the notion of an R − I − open set and obtained some of its properties. The purpose of this paper is to introduce a decomposition of continuity on F* – spaces and also to show that δI – r – continuity ( resp. pre – I – continuity, semi – δ – I – continuity, * – perfect continuity ) is equivalent to R – I – continuity ( resp. R – I – continuity, t – I – continuity, * – dense – in – itself continuity ) if the domain is an SA* – space. 2. PRELIMINARIES Let (X,τ) be a topological space, and A ⊂ X. Throughout this paper Cl(A) and Int(A) denote the closure and the interior of A with respect to τ, respectively. An ideal, I is defined as a nonempty collection of subsets of X satisfying the following two conditions: (1) If A∈I and B ⊂ A, then B∈I; (2) If A∈I and B∈I, then A∪B∈I. An ideal topological space is a topological space (X,τ) with an ideal I on X and is denoted by (X,τ,I). For a subset A of X, A* (I) = {x∈X  U∩A∉I for each neighborhood U of x} is called the local function of A with respect to I and τ (KESKİN et al. 2004). We simply write A* instead of A*(I) when there is no chance for confusion. Note that X* is often a proper subset of X. The hypothesis that X = X* (HATIR & NOIRI 2005) is equivalent to the hypothesis that τ ∩ I = Ø(Levine, 1963). The ideal topological spaces which satisfy this hypothesis are called Hayashi – Samuels space. (ANKOVIĆ & HAMLETT 1990). For every ideal topological space (X, τ, I), there exists a topology τ*(I), finer than τ, generated by β (I,τ) = {U \ I: U∈τ and I∈I}, but in general β (I,τ) is not always a topology (JANKOVIĆ & HAMLETT 1990). Additionally, Cl*(A) = A∪A* defines a Kuratowski closure operator for τ*(I). First we shall recall some definitions that will be used in the sequel. DEFINITION 1. A subset A of an ideal topological space (X, τ) is said to be regular open (DUGUNDJI 1966) ( semi – open (KURATOWSKI 1966)) if A = Int(Cl(A)) ( A⊂Cl(Int(A)) ). DEFINITION 2. A subset A of an ideal topological space (X, τ, I) is said to be a) α – I – open (HATIR & NOIRI 2002)if A ⊂ Int(Cl*(Int(A))), b) α*– I – set (HATIR & NOIRI 2002) if A = Int(Cl*(Int(A))), c) pre – I – open (DONTCHEV 1996) if A ⊂ Int(Cl*(A)), d) R – I – open (YUKSEL et. al. 2005) if A = Int(Cl*(A)), e) t – I – set (HATIR & NOIRI 2002) if Int(A) = Int(Cl*(A)), f) δ – I – open (ACİKGOZ et. al. 2004) if Int(Cl*(A)) ⊂ Cl*(Int(A)), g) regular I – closed (SAMUELS 1975) if A=(Int(A))*, h) I – open (ABD EL – MONSEF et. al. 1992) if A ⊂ Int((A)*), i) fI – set (KESKİN et. al. 2004) if A⊂(Int(A))*, j) semi – I – open (HATIR & NOIRI 2002) if A⊂Cl*(Int(A)), SDÜ FEN EDEBİYAT FAKÜLTESİ FEN DERGİSİ (E-DERGİ). 2008, 3(1), 51-59 53 k) β – I – open (HATIR & NOIRI 2002) if A⊂Cl(Int(Cl*(A))), l) * – perfect (HAYASHI 1964) if A=A*, m) * – dense – in – itself (HAYASHI 1964) if A⊂A*, n) I – locally closed (DONTCHEV 1999) if A=U∩V, where U is open and V is * – perfect, o) BI – set (HATIR & NOIRI 2002) if A=U∩V, where U is open and V is t – I – set, p) CI – set (HATIR & NOIRI 2002) if A=U∩V, where U is open and V is α*– I – set. The family of all R – I – open (resp. α – I – open, pre – I – open, t – I – set, δ – I – open, * – perfect set, * – dense – in – itself) sets in an ideal topological space (X, τ, I) is denoted by RIO (X,τ) ( resp. αIO (X,τ), PIO (X,τ), tIO (X,τ), δIO (X,τ), *PI (X,τ), *DI (X,τ) ). DEFINITION 3. A subset A of an ideal topological space (X, τ, I) is said to be δ – I – regular (ACIKGOZ &YUKSEL 2006) if A is both a pre – I – open set and a δ – I – open set. The family of all δ – I – regular sets of (X, τ, I) is denoted by δIR (X,τ), when there is no chance for confusion with the ideal. The following diagram is given by Acikgoz et al. (ACIKGOZ & YUKSEL 2006). regular open set DIAGRAM I R – I – open set t – I – set open set pre – I – open set δ – I – open set α* – I – set 3. ON F*– SPACES AND SA*– SPACES PROPOSITION 1. Let (X,τ,I) be an ideal topological space and A a subset of X. Then the following properties hold: a) If A is an R – I – open set and (X, τ, I) is a Hayashi-Samuels space, then A is an I – locally closed set, b) If A is an R – I – open set, then A is a BI – set. c) If A is a BI – set, then A is a CI – set. PROOF. a) Let A be an R – I – open set. Since (X,τ,I) is a Hayashi-Samuels space, then X* = X. Since every R – I – open set is an open set by (ACIKGOZ & YUKSEL 2006) and X is a * – perfect set, A = A∩X is an I – locally closed set. A. ACIKGOZ, S. YUKSEL, I. L. REILLY 54 b) Let A be an R – I – open set. Hence A is a t – I – set by (ACIKGOZ & YUKSEL 2006). Since X is an open set, A (...truncated)