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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 .
the minimal cardinality of a dense subset of a topological space. A set of density ℵ 0 is a separable space. [5] Derived set If X is a space and S is a subset of X, the derived set of S in X is the set of limit points of S in X. Developable space A topological space with a development. [6] Development
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 ...
Baire space. A space X is a Baire space if it is not meagre in itself. Equivalently, X is a Baire space if the intersection of countably many dense open sets is dense. Door space. A topological space is a door space if every subset is open or closed (or both). Topological Homogeneity.
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold: Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods.
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.