Editing First-countable space (section)

{{short description|Topological space where each point has a countable neighbourhood basis}}
In [[topology]], a branch of [[mathematics]], a '''first-countable space''' is a [[topological space]] satisfying the "first [[axiom of countability]]". Specifically, a space <math>X</math> is said to be first-countable if each point has a [[countable]] [[Neighbourhood system#Basis|neighbourhood basis]] (local base). That is, for each point <math>x</math> in <math>X</math> there exists a [[sequence]] <math>N_1, N_2, \ldots</math> of [[Neighbourhood (topology)|neighbourhoods]] of <math>x</math> such that for any neighbourhood <math>N</math> of <math>x</math> there exists an integer <math>i</math> with <math>N_i</math> [[Subset|contained in]] <math>N.</math>
Since every neighborhood of any point contains an [[open set|open]] neighborhood of that point, the [[Neighbourhood system|neighbourhood basis]] can be chosen [[without loss of generality]] to consist of open neighborhoods.