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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 .
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.
Compact space. Relatively compact subspace; Heine–Borel theorem; Tychonoff's theorem; Finite intersection property; Compactification; Measure of non-compactness; Paracompact space; Locally compact space; Compactly generated space; Axiom of countability; Sequential space; First-countable space; Second-countable space; Separable space ...
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
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 ...
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).
{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.
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 ...