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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.
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 ...
Entrance sign at the tunnels. Part of the tunnel complex at Củ Chu, this tunnel has been made wider and taller to accommodate tourists. The tunnels of Củ Chi (Vietnamese: Địa đạo Củ Chi) are an immense network of connecting tunnels located in the Củ Chi District of Ho Chi Minh City (Saigon), Vietnam, and are part of a much larger network of tunnels that underlie much of the country.
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 ...
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
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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