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A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), but every open subspace of a separable space is separable (Willard 1970, Th 16.4b). Also every subspace of a separable metric space is separable. In fact, every topological space is a subspace of a separable space of the same cardinality.
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
This space has cardinality c^c "A separable, Hausdorff space X has cardinality less than or equal to 2^c" why is IN^IN >= 2^c = c^c? It looks like that c is the cardinality of the reals. If c would be the cardinality of the natural numbers, I think it would be OK —Preceding unsigned comment added by 77.4.18.100 ( talk ) 09:01, 4 July 2008 ...
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
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A Hausdorff topological space is a Suslin space (named after Mikhail Suslin) if it is the image of a Polish space under a continuous mapping. So every Lusin space is Suslin. In a Polish space, a subset is a Suslin space if and only if it is a Suslin set (an image of the Suslin operation). [9] The following are Suslin spaces:
In a separable space, biorthogonality is not a substantial obstruction to a Markushevich basis; any spanning set and separating functionals can be made biorthogonal. But it is an open problem whether every separable Banach space admits a Markushevich basis with ‖ x α ‖ = ‖ f α ‖ = 1 {\displaystyle \|x_{\alpha }\|=\|f_{\alpha }\|=1 ...