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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 closed if and only if every convergent sequence in the set has its limit point 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
Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf. To further compare these two properties: An arbitrary subspace of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
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, separable, and Lindelöf. Every σ-compact space is Lindelöf. A metric space is first-countable. For metric spaces ...
A space is first-countable if every point has a countable local base. 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 ...
A metric space is σ-compact if and only if it is Lindelöf. [9] Every second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset). [8] A metric space is separable if and only if it is σ-compact. [9] Every sequentially continuous real-valued function in a metric space is a continuous ...
A separable metric space is completely metrizable if and only if the second player has a winning strategy in this game. A second characterization follows from Alexandrov's theorem. It states that a separable metric space is completely metrizable if and only if it is a G δ {\displaystyle G_{\delta }} subset of its completion in the original metric.
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 ...