nLab
first-countable space

First-countable spaces

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.

Generalizations

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.

propertiesimplications
metrisable: topology is induced by a metrica metric space has a σ \sigma -locally discrete base
second-countable: there is a countable base of the topology.Nagata-Smirnov metrization theorem
σ\sigma-locally discrete base, i.e. XX has a σ \sigma -locally discrete base.a second-countable space has a σ \sigma -locally finite base: take the the collection of singeltons of all elements of countable cover of XX.
σ\sigma-locally finite base, i.e. XX has a countably locally finite base.second-countable spaces are separable: choose a point in each set of countable cover.
separable: there is a countable dense subset.second-countable spaces are Lindelöf
Lindelöf: every open cover has a countable sub-cover.weakly Lindelöf spaces with countably locally finite base are second countable
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.separable metacompact spaces are Lindelöf
countable choice: the natural numbers is a projective object in Set.separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.
metacompact: every open cover has a point-finite open refinement.Lindelöf spaces are trivially also weakly Lindelöf.
first-countable: every point has a countable neighborhood basea space with a σ\sigma-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a σ\sigma-locally finite base.
Frechet-Uryson space: the closure of a set AA consists precisely of all limit points of sequences in AAa first-countable space is Fréchet-Urysohn: obvious
sequential topological space: a set AA is closed if it contains all limit points of sequences in AAobviously, a Fréchet-Uryson space is sequential
countably tight: for each subset AA and each point xA¯x\in \overline A there is a countable subset DAD\subseteq A such that xD¯x\in \overline D.obviously, a sequential space is countably tight.

Last revised on April 5, 2019 at 19:49:17. See the history of this page for a list of all contributions to it.