Editing Second-countable space (section)

==Properties==

Second-countability is a stronger notion than [[first-countable space|first-countability]]. A space is first-countable if each point has a countable [[local base]]. Given a base for a topology and a point ''x'', the set of all basis sets containing ''x'' forms a local base at ''x''. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable [[discrete space]] is first-countable but not second-countable.

Second-countability implies certain other topological properties. Specifically, every second-countable space is [[separable space|separable]] (has a countable [[dense (topology)|dense]] subset) and [[Lindelöf space|Lindelöf]] (every [[open cover]] has a countable subcover). The reverse implications do not hold. For example, the [[lower limit topology]] on the real line is first-countable, separable, and Lindelöf, but not second-countable. For [[metric space]]s, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.<ref>Willard, theorem 16.11, p. 112</ref>  Therefore, the lower limit topology on the real line is not metrizable.

In second-countable spaces&mdash;as in metric spaces&mdash;[[compact space|compactness]], sequential compactness, and countable compactness are all equivalent properties.

[[Urysohn's metrization theorem]] states that every second-countable, [[Hausdorff space|Hausdorff]] [[regular space]] is [[metrizable]]. It follows that every such space is [[completely normal space|completely normal]] as well as [[paracompact]]. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.

===Other properties===

*A continuous, [[open map|open]] [[image (mathematics)|image]] of a second-countable space is second-countable.
*Every [[subspace (topology)|subspace]] of a second-countable space is second-countable.
*[[Quotient space (topology)|Quotients]] of second-countable spaces need not be second-countable; however, ''open'' quotients always are.
*Any countable [[product space|product]] of a second-countable space is second-countable, although uncountable products need not be.
*The topology of a second-countable T<sub>1</sub> space has [[cardinality]] less than or equal to ''c'' (the [[cardinality of the continuum]]).
*Any base for a second-countable space has a countable subfamily which is still a base.
*Every collection of disjoint open sets in a second-countable space is countable.