First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space formula_1 is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point formula_2 in formula_1 there exists a sequence formula_4 of neighbourhoods of formula_2 such that for any neighbourhood formula_6 of formula_2 there exists an integer formula_8 with formula_9 contained in formula_10 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 formula_2 with radius formula_12 for integers form a countable local base at formula_13 An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line). Another counterexample is the ordinal space formula_14 where formula_15 is the first uncountable ordinal number. The element formula_15 is a limit point of the subset  formula_17 even though no sequence of elements in formula_17 has the element formula_15 as its limit. In particular, the point formula_15 in the space formula_14 does not have a countable local base. Since formula_15 is the only such point, however, the subspace formula_23 is first-countable. The quotient space formula_24 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 formula_25 and every element formula_2 in the closure of formula_27 there is a sequence in A converging to formula_13 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 formula_27 a point formula_2 lies in the closure of formula_25 if and only if there exists a sequence formula_32 in formula_25 which converges to formula_13 (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 formula_35 is a function on a first-countable space, then formula_35 has a limit formula_37 at the point formula_2 if and only if for every sequence formula_39 where formula_40 for all formula_41 we have formula_42 Also, if formula_35 is a function on a first-countable space, then formula_35 is continuous if and only if whenever formula_39 then formula_46 In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily non-metric spaces). One such space is  the ordinal space formula_47 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.