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Separation axioms in topological spaces
Kolmogorov classification
T0
(Kolmogorov)
T1
(Fréchet)
T2
(Hausdorff)
T2½
(Urysohn)
completely T2
(completely Hausdorff)
T3
(regular Hausdorff)
T3½
(Tychonoff)
T4
(normal Hausdorff)
T5
(completely normal Hausdorff)
T6
(perfectly normal Hausdorff)
History
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. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms.
^Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
topology and related fields of mathematics, a topological space X is called a regularspace if every closed subset C of X and a point p not contained...
Tychonoff spaces and completely regularspaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space is any...
Hausdorff. Every regular second-countable space is completely normal, and every regular Lindelöf space is normal. Also, all fully normal spaces are normal (even...
Regularspace (or T 3 {\displaystyle T_{3}} ) space, a topological space in which a point and a closed set can be separated by neighborhoods Regular army...
of the American animated comedy television series Regular Show (promoted as Regular Show in Space), created by J. G. Quintel, originally aired on Cartoon...
A regular expression (shortened as regex or regexp), sometimes referred to as rational expression, is a sequence of characters that specifies a match...
call locally compact. Spaces satisfying condition (4) are exactly the locally compact regularspaces. Indeed, such a space is regular, as every point has...
separated by a continuous function. Every completely Hausdorff space is also T2½. X is regular if, given any point x and closed set F in X such that x does...
diagonals (60 face diagonals, 100 space diagonals). It is represented by the Schläfli symbol {5,3}. If the edge length of a regular dodecahedron is a {\displaystyle...
In typography, a thin space is a space character whose width is usually 1⁄5 or 1⁄6 of an em. It is used to add a narrow space, such as between nested...
metrization theorem. This states that every Hausdorff second-countable regularspace is metrizable. So, for example, every second-countable manifold is metrizable...
class of regular languages. In practice, most nonregular problems are solved by machines taking at least logarithmic space. To locate the regular languages...
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank....
geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent...
inscribed inside all other four Platonic solids, and tessellate space alongside the regular octahedron in the alternated cubic honeycomb. The third dimension...
Regular Show (known as Regular Show in Space during its eighth and final season) is an American animated sitcom created by J. G. Quintel for Cartoon Network...
representation). As in the case of curves in two dimensions, the curvature of a regularspace curve C in three dimensions (and higher) is the magnitude of the acceleration...
figures are regular skew polygons, vertices zig-zagging between two planes. Finite regular skew polyhedra exist in 4-space. These finite regular skew polyhedra...
Spacey McSpaceTree (voiced by Steve Little) is a minor character in Regular Show. He made his debut in the Season Eight episodes "Spacey McSpaceTree" and...
space. The topology of a uniformizable space is always a symmetric topology; that is, the space is an R0-space. Conversely, each completely regular space...
convex regular polygon is denoted by its Schläfli symbol {n}. For n < 3, we have two degenerate cases: Monogon {1} Degenerate in ordinary space. (Most...
Waters of Eternity (1953) features hyperspace that allows observation of regularspace from within. Many stories feature hyperspace as a dangerous, treacherous...
In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable...
mathematics, a Hausdorff space (/ˈhaʊsdɔːrf/ HOWSS-dorf, /ˈhaʊzdɔːrf/ HOWZ-dorf), separated space or T2 space is a topological space where, for any two distinct...
respectively. A regular skew polyhedron has regular polygon faces, and a regular skew polygon vertex figure. Three infinite regular skew polyhedra are space-filling...