{{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 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 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> contained in <math>N.</math> Since every neighborhood of any point contains an open neighborhood of that point, the 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 balls centered at <math>x</math> with radius <math>2^{-n}</math> for all natural numbers <math>n </math> 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 <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 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 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 limits and 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, sequential compactness and 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 ordinal space <math>\left[0, \omega_1\right).</math> Every first-countable space is compactly generated.
Every subspace of a first-countable space is first-countable. Any countable 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