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.