~436~ ISSN Print: 2394-7500 ISSN Online: 2394-5869 Impact Factor: 5.2 IJAR 2016; 2(9): 436-442 www.allresearchjournal.com Received: 01-07-2016 Accepted: 02-08-2016 B Kanchana Research Scholar, Department of Mathematics, Nirmala College for Women, Coimbatore, India F Nirmala Irudayam Assistant Professor, Department of Mathematics, Nirmala College for Women, Coimbatore, India Correspondence B Kanchana Research Scholar, Department of Mathematics, Nirmala College for Women, Coimbatore, India New forms of continuous and closed maps via b-open sets in SETS B Kanchana and F Nirmala Irudayam Abstract In this paper we propose the concept of a new class of functions named strongly g *+ b-continuous, perfectly g *+ b-continuous, g *+ b-closed and open maps in simple extended topological spaces (SETS). Also we intend to define a new class of homeomorphism called g *+ b-homeomorphism in SETS. Some of their basic properties and several characterizations of these type of functions are discussed. Keywords: Strongly g *+ b-continuous, perfectly g *+ b-continuous, g *+ b-closed and open maps, g *+ b- homeomorphism 1. Introduction Levine [9] initiated the concept of generalized closed sets in topological spaces and a class of topological spaces called T1/2-spaces. The strong forms of continuous maps named strongly continuous maps, perfectly continuous maps, completely continuous maps and clopen continuous maps were introduced by Levine [10] , Noiri [13] , Munshi and Bassan [12] , Reilly and Vamanamurthy [15] respectively. Bharathi [2] studied the concept of strongly and perfectly g * b-continuous functions in topological spaces. Malghan [11] delivered the idea of generalized closed mappings in topological spaces. Biswas [1] , Sundaram [17] , Crossley and Hildebrand [3] have introduced and investigated semi-open maps and several generalized mappings in topological spaces and also discussed a class of homeomorphisms named semi homeomorphism, some what homeomorphism, generalized homeomorphism and gc- homeomorphism respectively. Vidhya and Parimelazhagan [20] devised the concept of g * b- closed (open) maps and g * b-homeomorphism in topological spaces. Levine [8] proposed the concept of extending a topology by a non-open set for τ is defined as τሺBሻ ൌ ሼሺB ∩ Oሻ ∪ O ᇱ / O, O ᇱ ∈ τሽ in 1963. B. Kanchana and F. Nirmala Irudayam [6, 7] formulated the concept of g *+ b-closed sets and g *+ b-continuity in extended topological spaces. The purpose of the present paper is to study some new forms of g *+ b-continuous, g *+ b-closed maps and homeomorphism in extended topological spaces and investigate some of their properties. Throughout this paper X, Y and Z (or (,  ା ), ሺ, ߪ ା ሻ and ሺ, ߟ ା ሻ) are simple extension topological space in which no separation axioms are assumed unless and otherwise stated. For any subset A of X, the interior of A is same as the interior in usual topology and the closure of A is newly defined in simple extension topological spaces. 2. Preliminaries We recall the following definitions which are useful in the sequel. Definition 2.1: A subset A of a topological space (,) is called a, (i) generalized b-closed set (briefly gb-closed) [14] , if bcl(A) ⊆ U, whenever A ⊆ U and U is open in X. (ii) g * b-closed set [18] , if bcl(A) ⊆ U, whenever A ⊆ U and U is g-open in X. Definition 2.2: A subset A of a topological space (, ା ) is called a, (i) generalized b + -closed set (briefly gb + -closed) [6] , if bcl + (A) ⊆ U, whenever A ⊆ U and U is open in X. (ii) g *+ b-closed set [6] , if bcl + (A) ⊆ U, whenever A ⊆ U and U is g + -open in X. International Journal of Applied Research 2016; 2(9): 436-442