Definition:Countability Axioms

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Definitions

Countability axioms is the common name used to refer to a set of properties of a topological space which have to do with the existence of countable sets, or countable families of open sets, satisfying certain conditions.

They are not axioms in the strict sense of the word, but they are usually named as such because one may think of them as additional basic properties that one can ask from a topological space.

Euclidean space $\R^N$ satisfies all of the axioms below, and the same happens for many usual spaces, so they are properties that one may rely upon in many situations.


First-Countable Space

A topological space $T = \left({S, \tau}\right)$ is first-countable or satisfies the First Axiom of Countability if and only if every point in $S$ has a countable local basis.


Second-Countable Space

A topological space $T = \left({S, \tau}\right)$ is second-countable or satisfies the Second Axiom of Countability if and only if its topology has a countable basis.


Separable Space

A topological space $T = \left({S, \tau}\right)$ is separable if and only if there exists a countable subset of $S$ which is everywhere dense in $T$.


Lindelöf Space

A topological space $T = \left({S, \tau}\right)$ is a Lindelöf space if and only if every open cover of $S$ has a countable subcover.


Also see