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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 {\displaystyle X} is said to be first-countable if each point has a countable neighbourhood basis (local base).
Important countability axioms for topological spaces include: [1] sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set; first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base
G is (Hausdorff and) first countable (equivalently: the identity element 1 is closed in G, and there is a countable basis of neighborhoods for 1 in G). G is metrisable (as a topological space). There is a left-invariant metric on G that induces the given topology on G. There is a right-invariant metric on G that induces the given topology on G.
Second-countable. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf. Lindelöf. A space is Lindelöf if every open cover has a countable subcover. σ-compact. A space is σ-compact if it is the union of countably many compact subspaces.
Cocountable topology. Given a topological space (,), the cocountable extension topology on is the topology having as a subbasis the union of τ and the family of all subsets of whose complements in are countable. Cofinite topology; Double-pointed cofinite topology
A space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space.
If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity.
Every locally compact group which is first-countable is metrisable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete. If furthermore the space is second-countable, the metric can be chosen to be proper. (See the article on topological groups.)