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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.
The name separated space is also used. A related, but weaker, notion is that of a preregular space. is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. A preregular space is also called an R 1 space. The relationship between these two conditions is as follows.
A topological space X is connected if these are the only two possibilities. Conversely, if a nonempty subset A is separated from its own complement, and if the only subset of A to share this property is the empty set, then A is an open-connected component of X .
Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as "the Hilbert space" or just "Hilbert space". [97] Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
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 .
The term symmetric space also has another meaning.) A topological space is a T 1 space if and only if it is both an R 0 space and a Kolmogorov (or T 0) space (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R 0 space if and only if its Kolmogorov quotient is a T 1 space.