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A first-countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuum cardinality. In such a space, closure is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset ...
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 .
Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation; Separable polynomial, a polynomial whose number of distinct roots is equal to its degree; Separable sigma algebra, a separable space in measure theory; Separable space, a topological space that contains a countable, dense subset
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
In the argument below denotes an infinite-dimensional separable Fréchet space and the relation of topological equivalence (existence of homeomorphism). A starting point of the proof of the Anderson–Kadec theorem is Kadec's proof that any infinite-dimensional separable Banach space is homeomorphic to R N . {\displaystyle \mathbb {R} ^{\mathbb ...
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 subset of its completion in the original metric.
If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly . If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to X , and so X has cardinality at least 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} .
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