### On (ω)compactness and (ω)paracompactness

Rupesh Tiwari, Manoj Kumar Bose
2012 Kyungpook Mathematical Journal
The notions of (ω)compactness and (ω)paracompactness are studied in product (ω)topology. The concept of countable (ω)paracompactness is introduced and some results on this notion are obtained. Definition 2.1(Bose and Tiwari ). If {J n } is a sequence of topologies on a set X with J n ⊂ J n+1 for all n ∈ N, then the pair (X, {J n }) is called an (ω)topological space. Definition 2.2(Bose and Tiwari ). Let (X, {J n }) be an (ω)topological space. A set G ∈ J n for some n is called an
more » ... called an Definition 2.3(Bose and Tiwari  ). An (ω)topological space X is said to be (ω)compact if every (ω)open cover of X has a finite subcover. Definition 2.4(Michael ). A collection A of a subsets of a topological space (X, J ) is said to be (J )locally finite if every x ∈ X has a (J )open neighbourhood intersecting finitely many sets A ∈ A. Definition 2.5(Bose and Tiwari ). An (ω)topological space (X, {J n }) is said to be (ω)paracompact if every (ω)open cover of X has, for some n, a (J n )locally finite (J n )open refinement. Definition 2.6(Dugundji , p.152). A collection A of subsets of a topological space (X, J ) is said to be point finite if for every x ∈ X, there exist at most finitely many sets A ∈ A such that x ∈ A. Theorem 2.7(Dugundji , p.152). A topological space (X, J ) is normal iff for every point finite open cover {U α |α ∈ A} of X, there exists a (J )open cover New definitions and theorems We introduce the following definition. Definition 3.1. Let (X, {A n }) and (Y, {B n }) be two (ω)topological spaces and for each n, P n be the product topology on X × Y of the topologies A n and B n on X and Y respectively. Then (X × Y, {P n }) is called the product (ω)topological space of the spaces (X, {A n }) and (Y, {B n }).