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36 relations: Base (topology), Clopen set, Closed set, Counterexample, Disjoint sets, Glossary of topology, Hausdorff space, History of the separation axioms, Inductive dimension, Interior (topology), Kolmogorov space, Locally compact space, Locally regular space, Mathematical analysis, Mathematics, Neighbourhood (mathematics), Neighbourhood system, Non-Hausdorff manifold, Normal space, Paracompact space, Point (geometry), Prentice Hall, Pseudonormal space, Semiregular space, Separated sets, Separation axiom, Subset, T1 space, Theorem, Topological indistinguishability, Topological space, Topology, Trivial topology, Tychonoff space, Urysohn and completely Hausdorff spaces, Zero-dimensional space.
Base (topology)
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.
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Clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule or law.
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Disjoint sets
In mathematics, two sets are said to be disjoint sets if they have no element in common.
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Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology.
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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History of the separation axioms
The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept.
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Inductive dimension
In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X).
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Interior (topology)
In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.
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Kolmogorov space
In 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 of distinct points of X, at least one of them has a neighborhood not containing the other.
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Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
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Locally regular space
In mathematics, particularly topology, a topological space X is locally regular if intuitively it looks locally like a regular space.
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Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
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Neighbourhood system
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x is the collection of all neighbourhoods for the point x.
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Non-Hausdorff manifold
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space.
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Normal space
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 disjoint open neighborhoods.
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Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.
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Point (geometry)
In modern mathematics, a point refers usually to an element of some set called a space.
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Prentice Hall
Prentice Hall is a major educational publisher owned by Pearson plc.
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Pseudonormal space
In mathematics, in the field of topology, a topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them.
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Semiregular space
A semiregular space is a topological space whose regular open sets (sets that equal the interiors of their closures) form a base.
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Separated sets
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching.
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Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other.
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Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.
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Topological indistinguishability
In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.
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Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.
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Urysohn and completely Hausdorff spaces
In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods.
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Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.
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Extension by continuity, Regular Hausdorff space, Regular topological space, T3 space, T3-separation axiom, T3-space, T₃ space.
References
[1] https://en.wikipedia.org/wiki/Regular_space
