axioms of countability

first-countable, second-countable
#topology

Definition

A1 (first-countable)

A first-countable topological space $(X, \mathcal{T})$ , (i.e. $A_1$ -space) is one with a countable neighborhood basis, i.e.

for any $x \in X$ , there exists a countable family of open neighborhoods of $x$ , $\{U_1^x, U_2^x, U_3^x, ...\}$ , such that for any $U$ open neighborhood of $x$ , there exists $n$ such that $U_n^x \subset U$

A2 (second-countable)

A second-countable topological space $(X, \mathcal{T})$ , (i.e. $A_2$ -space) is one with a countable basis i.e.

there exists a countable family of open sets {U1,U2,...}\{U_1, U_2,...\} which form a base of topology 𝒯\mathcal{T}
- i.e. any open set is a union of (not necessarily finite) sets in the family (countable subcollection)

See also

separation axioms, e.g. Hausdorff spaces

References