topological space

A topological spaceMathworldPlanetmath is a set X together with a set 𝒯 whose elements are subsets of X, such that

  • 𝒯

  • X𝒯

  • If Uj𝒯 for all jJ, then jJUj𝒯

  • If U𝒯 and V𝒯, then UV𝒯

Elements of 𝒯 are called open sets of X. The set 𝒯 is called a topologyMathworldPlanetmath on X. A subset CX is called a closed setPlanetmathPlanetmath if the complement XC is an open set.

A topology 𝒯 is said to be finer (respectively, coarserPlanetmathPlanetmath) than 𝒯 if 𝒯𝒯 (respectively, 𝒯𝒯).



  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title topological space
Canonical name TopologicalSpace
Date of creation 2013-03-22 11:49:52
Last modified on 2013-03-22 11:49:52
Owner djao (24)
Last modified by djao (24)
Numerical id 12
Author djao (24)
Entry type Definition
Classification msc 22-00
Classification msc 55-00
Classification msc 54-00
Synonym topology
Related topic NeighborhoodMathworldPlanetmathPlanetmath
Related topic MetricSpace
Related topic ExamplesOfCompactSpaces
Related topic ExamplesOfLocallyCompactAndNotLocallyCompactSpaces
Related topic Site
Defines open
Defines closed