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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 .
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).
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
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 subspace; Lindelöf space: every open cover has a countable subcover; σ-compact space: there exists a countable cover by compact spaces; Relations:
{p} is dense and hence X is a separable space. However if X is uncountable then X \ {p} is not separable. This is an example of a subspace of a separable space not being separable. Countability (first but not second) If X is uncountable then X is first countable but not second countable. Alexandrov-discrete The topology is an Alexandrov topology.
However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold is separable and paracompact. Moreover, if a manifold is separable and ...
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 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 ...