# Countability axiom

In the mathematical sub-area of topology, there are two finiteness conditions for the spaces under consideration , which are called the first and second axiom of countability . Spaces that satisfy an axiom of countability can be considered “small” from a topological point of view.

These two countability properties were introduced by Felix Hausdorff in his monograph, Basic Principles of Set Theory from 1914.

## First countability axiom

The first axiom of countability says:

Each point has a neighborhood base that can be counted at most .

That means: If a topological space and a point exist, then there is at most a countable set of neighborhoods of , so that for every neighborhood of there is an index such that it holds. A space that fulfills the first axiom of countability is called first countable. ${\ displaystyle X}$${\ displaystyle x \ in X}$${\ displaystyle \ {U_ {1}, U_ {2}, \ ldots \}}$${\ displaystyle x}$${\ displaystyle V}$${\ displaystyle x}$${\ displaystyle k}$${\ displaystyle U_ {k} \ subseteq V}$

### properties

The first axiom of countability is a local requirement; i.e., is an open covering of , so that the spaces with the subspace topology satisfy the first countability axiom, then the first countability axiom also holds for . ${\ displaystyle \ {V_ {i} \}}$${\ displaystyle X}$${\ displaystyle V_ {i}}$${\ displaystyle X}$

Convergent sequences are much less useful in spaces that do not satisfy the first countability axiom. For example, in such spaces a point of closure of a subset is not necessarily the limit of a sequence of elements . In order to describe closed sets by limit values, Moore-Smith sequences (networks) or filters must be considered in such spaces . ${\ displaystyle U}$${\ displaystyle U}$

## Second axiom of countability

The second axiom of countability says:

The space has at most a countable basis of the topology.

This means: If a topological space satisfies the second axiom of countability, then there is at most a countable set of open subsets which contains a neighborhood basis for every point, i.e. That is , for every point and every neighborhood of there is an index such that it holds. A space that satisfies the second axiom of countability is called second countable. ${\ displaystyle X}$${\ displaystyle \ {U_ {1}, U_ {2}, \ ldots \}}$${\ displaystyle x \ in X}$${\ displaystyle V}$${\ displaystyle x}$${\ displaystyle k}$${\ displaystyle x \ in U_ {k} \ subseteq V}$

### properties

• The second axiom of countability implies the first. In a topological space that satisfies the second axiom of countability, every open set can be represented as a (at most countable) union of sets from the base.
• Every second countable topological space is automatically separable ; that is, it has at most a countable dense subset . This can be constructed by selecting an element from each (non-empty) base set.
• The second axiom of countability carries over to arbitrary subsets, i. That is, every subset of a second-countable space becomes a second-countable topological space again with the induced topology . Note that subsets of separable spaces generally need not be separable.
• Countable products of second countable topological spaces are again second countable with regard to the product topology .
• Every second countable topological space is a Lindelöf space .
• A topological space is second countable if and only if it has a countable sub-base .

## Examples

• Every ( pseudo- ) metric space fulfills the first countability axiom, since for every point the -environments form a countable environment basis.${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon = 1.1 / 2.1 / 3, \ ldots}$
• A (pseudo-) metric space satisfies the second axiom of countability if and only if it is separable.
• The set of real numbers and all finite-dimensional real vector spaces with their usual topology (as normalized spaces ) fulfill both countability axioms, for example the spheres with rational center coordinates and rational diameter form a countable basis of the topology.
• Every discrete space satisfies the first axiom of countability, since every point has a neighborhood basis consisting of a single unitary set. An uncountable set with the discrete topology does not satisfy the second axiom of countability.
• A topological space with the indiscreet topology fulfills both axioms of countability.
• The Sorgefrey line satisfies the first countability axiom and is separable, but does not satisfy the second countability axiom.