# nLab first-countable space

First-countable spaces

# 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 $x$ has a countable local basis $B_x$.

## Generalizations

The character of a space at a point $x$ is the minimum of the cardinalities of the possible bases $B_x$. 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 $x$, 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 $\sigma$-locally discrete base
second-countable: there is a countable base of the topology.Nagata-Smirnov metrization theorem
$\sigma$-locally discrete base, i.e. $X$ has a $\sigma$-locally discrete base.a second-countable space has a $\sigma$-locally finite base: take the the collection of singeltons of all elements of countable cover of $X$.
$\sigma$-locally finite base, i.e. $X$ 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 $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base.
Frechet-Uryson space: the closure of a set $A$ consists precisely of all limit points of sequences in $A$a first-countable space is Fréchet-Urysohn: obvious
sequential topological space: a set $A$ is closed if it contains all limit points of sequences in $A$obviously, a Fréchet-Uryson space is sequential
countably tight: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$.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.