Topological space (nonfiction): Difference between revisions
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[[File:Topological_space_examples.png|thumb|Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.]]In [[Topology (nonfiction)|topology]] and related branches of [[Mathematics (nonfiction)|mathematics]], a '''topological space''' may be defined as a set of points, along with a set of neighborhoods for each point, satisfying a set of axioms relating points and neighborhoods. | [[File:Topological_space_examples.png|thumb|Four examples and two non-examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.]]In [[Topology (nonfiction)|topology]] and related branches of [[Mathematics (nonfiction)|mathematics]], a '''topological space''' may be defined as a set of points, along with a set of neighborhoods for each point, satisfying a set of [[Axiom (nonfiction)|axioms]] relating points and neighborhoods. | ||
The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. | The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. | ||
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== Nonfiction cross-reference == | == Nonfiction cross-reference == | ||
* [[Axiom (nonfiction)]] | |||
* [[Mathematics (nonfiction)]] | * [[Mathematics (nonfiction)]] | ||
* [[Topology (nonfiction)]] | * [[Topology (nonfiction)]] | ||
Revision as of 10:05, 6 August 2017
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighborhoods for each point, satisfying a set of axioms relating points and neighborhoods.
The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.
Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints.
Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.
The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
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Fiction cross-reference
Nonfiction cross-reference
External links:
- Topological space @ Wikipedia