M.TRINITA PRICILLA, I.AROCKIARANI / IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org ISSN : 2250-3021 Vol. 1, Issue 2, pp. 111-117
Some Stronger Forms of g¾ b –continuous Functions M.TRINITA PRICILLA * and I.AROCKIARANI * * *Department of Mathematics, Jansons Institute of Technology Karumathampatti, India **Department of Mathematics, Nirmala College for Women, Coimbatore – 641 046.
Abstract: The purpose of this paper is to introduce new classes of functions called strongly g¾ b –closed map, strongly g¾ b –continuous , perfectly g¾ b-continuous and strongly g¾ b –irresolute functions in supra topological spaces. Some properties and several characterizations of these types of functions are obtained. Also we investigate the relationship between these classes of functions.
1. Introduction In 1970,Levine [7] introduced the concept of generalized closed sets in topological spaces and a class of topological spaces called �1/2 spaces. Extensive research on generalizing closedness was done in recent years by many Mathematicians [4,5,7,8,9].Andrijevic[2] introduced a new class of generalized open sets in a topological space, the so-called b-open sets. In 1983, A.S.Mashhour et al [9] introduced the notion of supra topological spaces and studied S-S continuous functions and S* - continuous functions. In 2010, O.R.Sayed and Takashi Noiri [12] introduced supra b - open sets and supra b - continuity on topological spaces. In 2011,I.Arockiarani and M.Trinita Pricilla[3] introduced a new class of generalized b-open sets in supra topological spaces. In this paper we introduce and investigate notions of new classes of functions namely strongly g¾ –closed, strongly g¾b–closed , strongly g¾-continuous, strongly g¾b-continuous strongly g¾ –irresolute, strongly g¾ b – irresolute, almost g¾ –irresolute and almost g¾ b –irresolute functions in supra topological spaces. Relations between these types of functions and other classes of functions are obtained. We also note that the class of g¾b–closed map is properly placed between strongly g¾b–closed map and almost g¾b–closed map.
2. Preliminaries Definition: 2.1 [9] A subclass đ?œ? ∗ ďƒŒ đ?‘ƒ(đ?‘‹) is called a supra topology on X if X∈ đ?œ? ∗ and đ?œ? ∗ is closed under arbitrary union.(X, đ?œ? ∗ ) is called a supra topological space (or supra space).The members of đ?œ? ∗ are called supra open sets. Definition: 2.2 [9] The supra closure of a set A is defined as ClÂľ đ??´ =∊ đ??ľ: đ??ľ đ?‘–đ?‘ đ?‘ đ?‘˘đ?‘?đ?‘&#x;đ?‘Ž đ?‘?đ?‘™đ?‘œđ?‘ đ?‘’đ?‘‘ đ?‘Žđ?‘›đ?‘‘ đ??´ ⊆ đ??ľ The supra interior of a set A is defined as Int Âľ đ??´ =âˆŞ đ??ľ: đ??ľ đ?‘–đ?‘ đ?‘ đ?‘˘đ?‘?đ?‘&#x;đ?‘Ž đ?‘œđ?‘?đ?‘’đ?‘› đ?‘Žđ?‘›đ?‘‘ đ??´ ⊇ đ??ľ Definition 2.3 [12] Let (đ?‘‹,Âľ) be a supra topological space. A set A is called a supra b - open set if ďƒ? A ClÂľ (Int Âľ(A) ) ďƒˆ Int Âľ(Cl Âľ(A)) .The complement of a supra b - open set is called a supra b - closed set. Definition: 2.4 [3] Let (đ?‘‹,Âľ) be a supra topological space. A set A of X is called supra generalized - closed set (simply gÂľ closed) if clÂľ(A) ďƒ? U whenever A ďƒ? U and U is supra open. The complement of supra generalized - closed set is supra generalized - open set. Definition: 2.5 [3] Let (đ?‘‹,Âľ) be a supra topological space. A set A of X is called supra generalized b - closed set (simply gÂľ b closed) if bclÂľ(A) ďƒ? U whenever A ďƒ? U and U is supra open. The complement of supra generalized b - closed set is supra generalized b - open set. Definition: 2.6 [ 14] A function đ?‘“: đ?‘‹, đ?œ? → (đ?‘Œ, đ?œŽ) is said to be gÂľ b –continuous if đ?‘“ −1 (đ?‘‰) is gÂľ b - closed in đ?‘‹, đ?œ? for every supra closed set V of đ?‘Œ, đ?œŽ . Definition: 2.7 [14] A function đ?‘“: đ?‘‹, đ?œ? → (đ?‘Œ, đ?œŽ) is said to be gÂľ b –irresolute if đ?‘“ −1 (đ?‘‰) is gÂľ b - closed in đ?‘‹, đ?œ? for every gÂľ b - closed set V of đ?‘Œ, đ?œŽ .
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