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Most " nice" 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.

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 form 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 LindelöfIn mathematics, a Lindelof space is a topological space in which every open cover has a countable subcover. It is named for Ernst Leonard Lindelof. A Lindelof space is a generalization of the more commonly used notion of compactness which requires that th (every open cover has a countable subcover). The reverse implications do not hold. For example, the lower limit topologyIn mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. It is the topology generated by t on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaceIn mathematics, a metric space is a set (or "space") where a distance between points is defined. History Maurice Frechet introduced metric spaces in his work Sur quelques points du calcul fonctionnel Rendic. Palermo 22(1906) 1-74. Formal definition Formals, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.

In second-countable spaces—as in metric spaces— compactnessIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e, sequential compactness, and countable compactness are all equivalent properties.

Every regularIn ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. The term regular also refers to: In the military, a regular unit is a military unit that is second-countable space is actually completely normal as well as paracompact (since every regular Lindelöf space is paracompact).

Other properties:

• Every subspace of a second-countable space is second countable
• Every collection of disjoint open sets in a second-countable space is countable
• Any countable product of a second-countable space is second countable, although uncountable products need not be.
• Quotients of second-countable spaces need not be second countable.