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| In [[topology]], a branch of [[mathematics]], a '''first-countable space''' is a [[topological space]] satisfying the "first [[axiom of countability]]". Specifically, a space ''X'' is said to be first-countable if each point has a [[countable]] [[neighbourhood system|neighbourhood basis]] (local base). That is, for each point ''x'' in ''X'' there exists a [[sequence]] ''N''<sub>1</sub>, ''N''<sub>2</sub>, … of [[neighbourhood (topology)|neighbourhoods]] of ''x'' such that for any neighbourhood ''N'' of ''x'' there exists an integer ''i'' with ''N''<sub>''i''</sub> [[subset|contained in]] ''N''.
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| Since every neighborhood of any point is contained in an open neighborhood of that point the [[neighbourhood system|neighbourhood basis]] can be chosen w.l.o.g. to consist of open neighborhoods.
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| ==Examples and counterexamples==
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| 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 ''x'' with radius 1/''n'' for integers ''n'' > 0 form a countable local base at ''x''.
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| An example of a space which is not first-countable is the [[cofinite topology]] on an uncountable set (such as the [[real line]]).
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| Another counterexample is the [[ordinal space]] ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] where ω<sub>1</sub> is the [[first uncountable ordinal]] number. The element ω<sub>1</sub> is a [[limit point]] of the subset <nowiki>[</nowiki>0,ω<sub>1</sub>) even though no sequence of elements in <nowiki>[</nowiki>0,ω<sub>1</sub>) has the element ω<sub>1</sub> as its limit. In particular, the point ω<sub>1</sub> in the space ω<sub>1</sub>+1 = [0,ω<sub>1</sub>] does not have a countable local base. Since ω<sub>1</sub> is the only such point, however, the subspace ω<sub>1</sub> = <nowiki>[</nowiki>0,ω<sub>1</sub>) is first-countable.
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| The [[quotient space]] <math>\mathbb{R}/\mathbb{N}</math> where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a [[Sequential space#Fréchet–Urysohn space|Fréchet-Urysohn space]].
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| 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.
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| ==Properties==
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| One of the most important properties of first-countable spaces is that given a subset ''A'', a point ''x'' lies in the [[closure (topology)|closure]] of ''A'' if and only if there exists a [[sequence]] {''x''<sub>''n''</sub>} in ''A'' which [[limit of a sequence|converges]] to ''x''. This has consequences for [[limit of a function|limits]] and [[continuity (topology)|continuity]]. In particular, if ''f'' is a function on a first-countable space, then ''f'' has a limit ''L'' at the point ''x'' if and only if for every sequence ''x''<sub>''n''</sub> → ''x'', where ''x''<sub>''n''</sub> ≠ ''x'' for all ''n'', we have ''f''(''x''<sub>''n''</sub>) → ''L''. Also, if ''f'' is a function on a first-countable space, then ''f'' is continuous if and only if whenever ''x''<sub>''n''</sub> → ''x'', then ''f''(''x''<sub>''n''</sub>) → ''f''(''x'').
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| 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 which aren't compact (these are necessarily non-metric spaces). One such space is the [[Order topology|ordinal space]] <nowiki>[</nowiki>0,ω<sub>1</sub>). Every first-countable space is [[compactly generated space|compactly generated]].
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| 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.
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| ==See also==
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| *[[Second-countable space]]
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| *[[Separable space]]
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| ==References==
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| *{{Springer|id=f/f040430|title=first axiom of countability}}
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| {{DEFAULTSORT:First-Countable Space}}
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| [[Category:General topology]]
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| [[Category:Properties of topological spaces]]
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