Editing First-countable space (section)

== Properties ==

One of the most important properties of first-countable spaces is that given a subset <math>A,</math> a point <math>x</math> lies in the [[Closure (topology)|closure]] of <math>A</math> if and only if there exists a [[sequence]] <math>\left(x_n\right)_{n=1}^{\infty}</math> in <math>A</math> that [[Limit of a sequence|converges]] to <math>x.</math> (In other words, every first-countable space is a [[Fréchet-Urysohn space]] and thus also a [[sequential space]].) This has consequences for [[Limit of a function|limits]] and [[Continuity (topology)|continuity]]. In particular, if <math>f</math> is a function on a first-countable space, then <math>f</math> has a limit <math>L</math> at the point <math>x</math> if and only if for every sequence <math>x_n \to x,</math> where <math>x_n \neq x</math> for all <math>n,</math> we have <math>f\left(x_n\right) \to L.</math> Also, if <math>f</math> is a function on a first-countable space, then <math>f</math> is continuous if and only if whenever <math>x_n \to x,</math> then <math>f\left(x_n\right) \to f(x).</math>

In first-countable spaces, [[Sequentially compact space|sequential compactness]] and [[Countably compact space|countable compactness]] are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces that are not compact (these are necessarily not metrizable spaces).  One such space is the [[Order topology|ordinal space]] <math>\left[0, \omega_1\right).</math> Every first-countable space is [[Compactly generated space|compactly generated]].

Every [[Subspace (topology)|subspace]] of a first-countable space is first-countable. Any countable [[Product space|product]] of a first-countable space is first-countable, although uncountable products need not be.