Second-countable space

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In topology, a second-countable space is a topological space satisfying the "second axiom of countability". Specifically, a space is said to be second-countable if its topology has a countable base. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. In general, the finer the topology, the less likely it is to be second-countable.

Most "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rⁿ) with its usual topology is second-countable. Although the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a base.

[edit] Properties

Second-countability is a stronger notion than first-countability. Recall that 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 clearly has a countable local base at every point.

Second-countability implies certain other topological properties. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover). The reverse implications do not hold. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.

In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties.

Urysohn's metrization theorem states that every second-countable, regular Hausdorff space is metrizable. It follows that every such space is completely normal as well as paracompact. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.

Other properties:

A continuous, open image of a second-countable space is second-countable.
Every subspace of a second-countable space is second-countable.
Quotients of second-countable spaces need not be second countable; however, open quotients always are.
Any countable product of a second-countable space is second-countable, although uncountable products need not be.
The topology of a second-countable space has cardinality less than or equal to c (the cardinality of the continuum).
Any base for a second-countable space has a countable subfamily which is still a base.
Every collection of disjoint open sets in a second-countable space is countable.