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 X is said to be first-countable if each point has a countable neighbourhood basis (local base). That is, for each point x in X there exists a sequence N₁, N₂, … of neighbourhoods of x such that for any neighbourhood N of x there exists an integer i with N_icontained in N. 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.

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 x with radius 1/n for integers n > 0 form a countable local base at x.

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 ω₁+1 = [0,ω₁] where ω₁ is the first uncountable ordinal number. The element ω₁ is a limit point of the subset [0,ω₁) even though no sequence of elements in [0,ω₁) has the element ω₁ as its limit. In particular, the point ω₁ in the space ω₁+1 = [0,ω₁] does not have a countable local base. Since ω₁ is the only such point, however, the subspace ω₁ = [0,ω₁) is first-countable.

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