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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 .
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
is a Hausdorff space if any two distinct points in are separated by neighbourhoods. This condition is the third separation axiom (after T 0 and T 1), which is why Hausdorff spaces are also called T 2 spaces. The name separated space is also used. A related, but weaker, notion is that of a preregular space.
The existence of a line separating the two types of points means that the data is linearly separable In Euclidean geometry , linear separability is a property of two sets of points . This is most easily visualized in two dimensions (the Euclidean plane ) by thinking of one set of points as being colored blue and the other set of points as being ...
sequential space: a set is closed if and only if every convergent sequence in the set has its limit point 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
Every second-countable space is separable. A metric space is separable if and only if it is second-countable and if and only if it is Lindelöf. Clearly a MS is a space so if separable iff second countable; so should the second one not be In a Metric space the following are equivalent: -- space 2nd countable -- space separable -- space Lindelof
In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are multipartite quantum states that can be written as a tensor product of states in each space. The physical intuition behind these definitions is that product states have no correlation between ...