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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 .
Let be a countable basis of .Consider an open cover, =.To get prepared for the following deduction, we define two sets for convenience, := {:}, ′:=. A straight-forward but essential observation is that, = which is from the definition of base. [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; separable space: there exists a countable dense subset
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 ...
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 ...
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.
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.
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