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First-countable SpaceIn 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. Related topics
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