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A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. [1][2] Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch ...
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology.
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.
For every topological space Y, the projection is a closed mapping [11] (see proper map). Every open cover linearly ordered by subset inclusion contains X. [12] Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above). [13]
Topological vector space. In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations ...
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.
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (,) is said to be metrizable if there is a metric : [,) such that the topology induced by d is . Metrization theorems are theorems that give sufficient conditions for a topological space to ...
In a first-countable space (such as a metric space), is the set of all limits of all convergent sequences of points in . For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter" (as described in the article on filters in topology).