Copyright © 2018 Authors. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. International Journal of Engineering & Technology, 7 (4.10) (2018) 407-409 International Journal of Engineering & Technology Website: www.sciencepubco.com/index.php/IJET Research paper ∆ * -Locally Continuous Functions and ∆ * -Locally Irresolute Maps in Topological Spaces K.Meena 1 & K.Sivakamasundari 2 1 Department of Mathematics, Kumaraguru College of Technology, Coimbatore, TamilNadu, India. 2 Department of Mathematics, Avinashilingam University, Coimbatore, TamilNadu, India. *Corresponding author E-mail: meenarajarajan@yahoo.in Abstract The objective of the paper is to introduce a new types of continuous maps and irresolute functions called ∆ * -locally continuous functions and ∆ * -irresolute maps in topological spaces. The comparative study between these functions with other existing maps is discussed in this paper. Some significant results are also proved as an application of new spaces namely, ∆ * -submaximal space and   T * – space. Further the characteristics of these maps under composition maps are exhibited. Keywords: ∆ * -closed set , ∆ * -continuous functions , ∆ * -irresolute maps ,   T * -space and ∆ * -submaximal space. 1. Introduction The locally closed sets idea in a topological space was pioneered by Bourbaki [2]. Initially it was defined as the intersection of an open set and a closed set. Using locally closed sets, Ganster and Reilly [3] defined locally continuous functions and locally irresolute maps. Further the concepts of generalized locally closed (briefly, glc) sets, generalised locally continuous and generalised locally irresolute maps were introduced by Balachandran et.al.,[1]. Since then several topologists contributed their study to the development of generalizations of locally closed sets and locally continuous maps. In this paper, two types of new generalised maps namely, ∆ ∗ -locally continuous and ∆ ∗ -locally irresolute maps are defined. Their properties using   T * -space and ∆ ∗ -submaximal space are analysed in this paper. Some important results under composition of mappings are also analyzed in this paper. Remark: The notation δ(δg) ∗ was used instead of the notation ∆ ∗ -closed sets at the time of introducing the definition.[4] 2. Preliminaries The notations (M, µ), (N, ν) and (T, λ) represent topological spac- es where µ, ν and λ denote topologies defined on the non empty sets M, N and T respectively. Definition 2.1 A subset S of a topological space (M, μ) is called a i) ∆ * -closed set if δcl(S) ⊆ B whenever S ⊆ B where B is δg-open in (M, μ). The complement of a ∆ * -closed set is called ∆ * -open set. [4] ii) δ-open set if it is the union of regular open sets. The comple- ment of a δ-open set is called a δ-closed set.[9] Definition 2.2 A mapping g : (M, μ) ⟶ (Y, ν) is said to be i) δ-continuous if the inverse image of every closed set in (Y, ν) is δ-closed in (M, μ). [8] ii) ∆ * -continuous if the inverse image of every closed set in (Y, ν) is ∆ * -closed in (M, μ). [5] iii) ∆ * -irresolute if the inverse image of every ∆ * -open set in (Y, ν) is ∆ * -open in (M, μ). [6] iv) Contra ∆ * -irresolute if the inverse image of every ∆ * -closed set in (Y, ν) is ∆ * -open in (M, μ). [6] Definition 2.3 A subset S of a topological space (M, μ) is called a [7] i) ∆ * -locally closed set which is denoted as ∆ * lc-set if there exists a ∆ * -open set B and a ∆ * -closed set D of (M, μ) such that S = B ∩ D. ii) ∆ * lc * -set if there exists a ∆ * -open set B and a δ-closed set D of (M, μ) such that S = B ∩ D. iii) ∆ * lc ** -set if there exists a δ-open set B and a ∆ * -closed set D of (M, μ) such that S = B ∩ D. The collection of all ∆ * lc ( resp., ∆ * lc * -sets, ∆ * lc ** ) sets of (M, μ) is denoted by ∆ * LC(M, μ) (resp., ∆ * LC * (M, μ), ∆ * LC ** (M, μ) ). Remark 2.4 For a topological space (M, μ), the following inclusions are true. [7] a) δLC(M, μ) ⊆ ∆ * LC(M, μ) b) δLC(M, μ) ⊆ ∆ * LC ** (M, μ) ⊆ ∆ * LC(M, μ) c) δLC(M, μ) ⊆ ∆ * LC * (M, μ) ⊆ ∆ * LC(M, μ) Remark 2.5 Let g : (M, μ) ⟶ (Y, ν) be a ∆ * -irresolute map. Then the follow- ing statements are true. [7]