nLab
first-countable space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

First-countable spaces

Idea

A space (such as a topological space) is first-countable if, in a certain sense, there is only a countable amount of information locally in its topology. (Change ‘locally’ to ‘globally’ to get a second-countable space.)

Definitions

Definition

A topological space is first-countable if every point xx has a countable local basis B xB_x.

Generalisations

The character of a space at a point xx is the minimum of the cardinalities of the possible bases B xB_x. We are implicitly using the axiom of choice here, to suppose that this set of cardinalities (which really is a small set because bounded above by the number of neighbourhoods of xx, and inhabited by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.

Then a first-countable space is simply one whose characters are all countable.

The character, tout court, of a space is the supremum of the characters of its points; then a first-countable space is simply one with a countable character.

Examples

Any second-countable space must also be first-countable.

Any metric space is first-countable.

Created on September 8, 2012 08:35:00 by Toby Bartels (98.23.143.147)