Editing First-countable space

{{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.

== 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.

== 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.

== See also ==

* {{annotated link|Fréchet–Urysohn space}}
* {{annotated link|Second-countable space}}
* {{annotated link|Separable space}}
* {{annotated link|Sequential space}}

== References ==
{{reflist}}

== Bibliography ==

* {{Springer|id=f/f040430|title=first axiom of countability}}
* {{cite book | last = Engelking | first = Ryszard | authorlink=Ryszard Engelking | title=General Topology | publisher=Heldermann Verlag, Berlin | year=1989 | isbn=3885380064| edition = Revised and completed | series = Sigma Series in Pure Mathematics, Vol. 6}}

{{DEFAULTSORT:First-Countable Space}}
[[Category:General topology]]
[[Category:Properties of topological spaces]]