International Journal of Mathematical Archive-5(11), 2014, 106-113 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 5(11), Nov. – 2014 106 EXTENDED TOPOLOGICAL PROPERTIES ON MEASURE MANIFOLD S. C. P. Halakatti* 1 1 Department of Mathematics, Karnatak University Dharwad, Karnataka, India. H. G. Haloli 2 2 Research Scholar, Department of Mathematics, Karnatak University, Dharwad, Karnataka, India. (Received On: 12-11-14; Revised & Accepted On: 20-11-14) ABSTRACT In this paper we study extended topological properties like extended Heine-Borel property (eHBP), extended countable compactness and extended Lindelof property on a measure manifold (M,  1 , Σ 1 ,  1 ). Also their inter-relationship on (M,  1 , Σ 1 ,  1 ). Keywords: Extended Heine-Borel property, Extended Countable Compact and Extended Lindelof property, Measure Manifold. Subject Classification: 54-XX, 58-XX, 57NXX, 49Q15, 58AXX. 1. INTRODUCTION In paper [1] S.C.P Halakatti has developed the concept of measure manifold by modelling a non empty set M on a measure space (ℝ n , , Σ, ) In paper [4], same author investigated the possibility of generating a broader class of measure manifolds by employing pull back function and existence of inverse function theorem on a measure space (ℝ n , , Σ, ). The existence of a C ∞ , measurable homeomorphism and measure invariant function generates a measure manifold. In the present paper we study extended topological properties like eHBP, e-countable compactness, e - Lindelof on a measure manifold with the help of a C ∞ measurable homeomorphism and measure invariant map :(ℝ m ,  1 , Σ 1 ,  1 ) → (ℝ n , , Σ, ) and the inter - relationship between these properties. 2. PRELIMINARIES To generate a Measure Manifold we use some basic concepts of mathematical framework which has been developed in our papers [1], [2], [3] and [4]. Definition 2.1: Measurable Chart Let (U,  1 U � ,  1  � ) ⊆ (M,  1 , Σ 1 ) be a non empty measurable subspace of (M,  1 , Σ 1 ) if there exists a map ϕ : (U,  1 U � ,  1  � ) →  (U,  1 U � ,  1  � ) ⊆ (R n , , Σ), satisfying following conditions: (i)  is homeomorphism (ii) ϕ is measurable i.e.,  -1 (V) ∈ (M,  1 , Σ 1 )for every V ∈ (R n , , Σ) on (U,  1 U � ,  1  � ) ⊂ (M,  1 , Σ 1 ) then the structure ((U,  1 U � ,  1  � ), ) is called a measurable chart. Definition 2.2: Measure Chart A measurable chart ((U,  1 U � , Σ 1 U � ), ϕ) equipped with a measure μ 1 U � is called a measure chart, denoted by ((U,  1 U � , Σ 1 U � ), ϕ) satisfying following conditions: (i).  is homomorphism (ii).  is measurable if for every measurable subset V∈(R n , , Σ),  -1 (V) ∈ (M,  1 , Σ 1 ) is also measurable (iii).  is measurable invariant Then the structure ((U,  1 U � , Σ 1 U � , μ 1 U � ),ϕ) is called a measure chart. Corresponding Author: S. C. P. Halakatti* 1 1 Department of Mathematics, Karnatak University Dharwad, Karnataka, India.