Chapter 5 Countability and Separation

  • Published 2014

Abstract

Definition 5.1.1. A topological space (X, J ) is said to have a countable local basis (or countable basis) at a point x ∈ X if there exists a countable collection say Bx of open sets containing x such that for each open set U containing x there exists V ∈ Bx with V ⊆ U . Definition 5.1.2. A topological space (X, J ) is said to be first countable or said to satisfy the first countability axiom if for each x ∈ X there exists a countable local base at x. Examples 5.1.3. (i) Let (X, d) be a metric space then for each x ∈ X, Bx = {B(x, 1 n) : n ∈ N} is a countable local basis at x. Hence (X, Jd) is a first countable space. So, we say that every metric space (X, d) is a first countable space. (ii) Let X = N and J = {φ,X, {1}, {1, 2}, . . . , {1, 2, . . . , n}, . . . , } then obviously (X,J ) is a first countable topological space.

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@inproceedings{2014Chapter5C, title={Chapter 5 Countability and Separation}, author={}, year={2014} }