Finite topological space information

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...

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric...

agree in a metric space, but may not be equivalent in other topological spaces. One such generalization is that a topological space is sequentially compact...

mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by...

mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , τ ) {\displaystyle (X...

is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X {\displaystyle X} is...

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have...

In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time,...

Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds...

functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector...

finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces. Due to the fact that inverse images commute...

T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one...

metric spaces is Baire. Every locally compact sober space is a Baire space. Every finite topological space is a Baire space (because a finite space has only...

and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms...

In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X...

topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair...

claimed to be homeomorphic to the topological quotient. Goreham, Anthony. Sequential convergence in Topological Spaces Archived 2011-06-04 at the Wayback...

for infinite dimensional vector spaces. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same...

can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space). The term precompact...

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous...

space" and "topological dual space" are often replaced by "dual space". For a topological vector space V {\displaystyle V} its continuous dual space,...

convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can...

class of metrizable spaces. Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense...

mathematics Finite geometry Finite group, Finite ring, Finite field Finite topological space Duncan Luce (1957) American Mathematical Monthly 64:688 Mathematics...

In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition...

a topological space X {\displaystyle X} is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many...