Editing First-countable space (section)

== Examples and counterexamples ==

The majority of 'everyday' spaces in [[mathematics]] are first-countable. In particular, every [[metric space]] is first-countable. To see this, note that the set of [[open ball]]s centered at <math>x</math> with radius <math>1/n</math> for integers  form a countable local base at <math>x.</math>

An example of a space that is not first-countable is the [[cofinite topology]] on an [[uncountable set]] (such as the [[real line]]). More generally, the [[Zariski topology]] on an [[algebraic variety]] over an uncountable field is not first-countable.

Another counterexample is the [[ordinal space]] <math>\omega_1 + 1 = \left[0, \omega_1\right]</math> where <math>\omega_1</math> is the [[first uncountable ordinal]] number. The element <math>\omega_1</math> is a [[limit point]] of the subset <math>\left[0, \omega_1\right)</math> even though no sequence of elements in <math>\left[0, \omega_1\right)</math> has the element <math>\omega_1</math> as its limit. In particular, the point <math>\omega_1</math> in the space <math>\omega_1 + 1 = \left[0, \omega_1\right]</math> does not have a countable local base. Since <math>\omega_1</math> is the only such point, however, the subspace <math>\omega_1 = \left[0, \omega_1\right)</math> is first-countable.

The [[Quotient space (topology)|quotient space]] <math>\R / \N</math> where the natural numbers on the real line are identified as a single point is not first countable.<ref>{{Harv|Engelking|1989|loc=Example 1.6.18}}</ref> However, this space has the property that for any subset <math>A</math> and every element <math>x</math> in the closure of <math>A,</math> there is a sequence in <math>A</math> converging to <math>x.</math> A space with this sequence property is sometimes called a [[Fréchet–Urysohn space]].

First-countability is strictly weaker than [[second-countability]]. Every [[second-countable space]] is first-countable, but any uncountable [[discrete space]] is first-countable but not second-countable.