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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 .
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 ...
A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space. Polyadic A space is polyadic if it is the continuous image of the power of a one-point compactification of a locally compact, non-compact Hausdorff space. Polytopological space
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
Every first-countable space is sequential. Every second-countable space is first countable, separable, and Lindelöf. Every σ-compact space is Lindelöf. Every metric space is first countable. For metric spaces, second-countability, separability, and the Lindelöf property are all equivalent.
The conjecture is true if the Hilbert space is not separable (i.e. if it has an uncountable orthonormal basis). In fact, if x {\displaystyle x} is a non-zero vector in H {\displaystyle H} , the norm closure of the linear orbit [ x ] {\displaystyle [x]} is separable (by construction) and hence a proper subspace and also invariant.
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