First-countable space
In topology, 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 local base. That is, for each x ∈ X there exists a sequence U1, U2, … of open neighborhoods of x such that for any neighborhood V there exists an integer i with Ui ⊆ V.
Examples and counterexamples
Most "nice" 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 [0,ω1] where ω1 is the smallest 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, ω1 does not have a countable local base. The subspace [0,ω1) is first-countable however, since ω1 is the only such point.
Properties
One of the most important properties of first-countable spaces is that given a subset A, a point x lies in the closure of A if and only if there exists a sequence {xn} in A which converges to x.
In first-countable spaces, sequential compactness and 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 ordinal space [0,ω1). 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.