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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 ...
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.
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.
In topology and related branches of mathematics, a Hausdorff space (/ ˈhaʊsdɔːrf / HOWSS-dorf, / ˈhaʊzdɔːrf / HOWZ-dorf[1]), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T 2) is ...
Regular space. In topology and related fields of mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C have non-overlapping open neighborhoods. [1] Thus p and C can be separated by neighborhoods. This condition is known as Axiom T3.
Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space is a connected set if it is a connected space when viewed as a subspace of . Some related but stronger conditions are path connected, simply connected, and -connected.
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 an union of elements of some subfamily of .
Finite topological space. In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding ...