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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
Any second-countable space is separable: if {} is a countable base, choosing any from the non-empty gives 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.
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 ...
Every Polish space is second countable (by virtue of being separable and metrizable). [1]A subspace Q of a Polish space P is Polish (under the induced topology) if and only if Q is the intersection of a sequence of open subsets of P (i. e., Q is a G δ-set).
A compact metric space (X, d) also satisfies the following properties: Lebesgue's number lemma: For every open cover of X, there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (X, d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric ...
{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.
The sets and are separated by a continuous function if there exists a continuous function: from the space to the real line such that () and (), that is, members of map to 0 and members of map to 1. (Sometimes the unit interval [ 0 , 1 ] {\displaystyle [0,1]} is used in place of R {\displaystyle \mathbb {R} } in this definition, but this makes ...