Editing Second-countable space (section)

{{short description|Topological space whose topology has a countable base}}

In [[topology]], a '''second-countable space''', also called a '''completely separable space''', is a [[topological space]] whose topology has a [[countable]] [[base (topology)|base]]. More explicitly, a topological space <math>T</math> is second-countable if there exists some countable collection <math>\mathcal{U} = \{U_i\}_{i=1}^{\infty}</math> of [[open set|open]] subsets of <math>T</math> such that any open subset of <math>T</math> can be written as a union of elements of some subfamily of <math>\mathcal{U}</math>. A second-countable space is said to satisfy the '''second axiom of countability'''. Like other [[countability axiom]]s, the property of being second-countable restricts the number of open subsets that a space can have. 

Many "[[well-behaved]]" spaces in [[mathematics]] are second-countable. For example, [[Euclidean space]] ('''R'''<sup>''n''</sup>) with its usual topology is second-countable. Although the usual base of [[open ball]]s is [[uncountable]], one can restrict this to the collection of all open balls with [[rational number|rational]] radii and whose centers have rational coordinates. This restricted collection is countable and still forms a basis.