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In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.More explicitly, a topological space is second-countable if there exists some countable collection = {} = of open subsets of such that any open subset of can be written as a union of elements of some subfamily of .
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; separable space: there exists a countable dense subset
Every second-countable space is first-countable, separable, and Lindelöf. Semilocally simply connected A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simply connected space and every locally simply connected ...
second-countable space: the topology has a countable base; separable space: there exists a countable dense subspace; Lindelöf space: every open cover has a countable subcover; σ-compact space: there exists a countable cover by compact spaces; Relations: Every first countable space is sequential. Every second-countable space is first-countable ...
G is a second countable locally compact (Hausdorff) space. G is a Polish, locally compact (Hausdorff) space. G is properly metrisable (as a topological space). There is a left-invariant, proper metric on G that induces the given topology on G. Note: As with the rest of the article we of assume here a Hausdorff topology.
Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover (Kelley 1955:49). This means that every second-countable space is also a Lindelöf space.
Every subspace of a G δ space is a G δ space. Every metrizable space is a G δ space. The same holds for pseudometrizable spaces. Every second countable regular space is a G δ space. This follows from the Urysohn's metrization theorem in the Hausdorff case, but can easily be shown directly. [3] Every countable regular space is a G δ space.
A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf, [5] but not conversely. For example, there are many compact spaces that are not second-countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. [6] Every regular Lindelöf space ...