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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
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 ...
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 ...
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).
When w(X ) the space X is said to be second countable. The -weight of a space X is the smallest ... [0, 1] is a compact Hausdorff space, which is separable and so ccc.