Topological property

Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms.Informally, a topological property is a property of the space that can be expressed using open sets.A common problem in topology is to decide whether two topological spaces are homeomorphic or not.To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.is: Some of these terms are defined differently in older mathematical literature; see history of the separation axioms.For example, the metric space properties of boundedness and completeness are not topological properties.[2] Simon Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013).
topologymathematicstopological spaceinvarianthomeomorphismsproper classopen setshomeomorphicsubspacecardinalitybasis of the topologySeparation axiomhistory of the separation axiomsKolmogorovFréchetHausdorffUrysohncompletely T2separated by a functionregularregular Hausdorffconsistentcompletely regularTychonoff spacenormalpartitions of unitycompletely normalseparated setsperfectly normalprecisely separated by a functionperfectly normal Hausdorffdiscreteisolated pointsAxiom of countabilityseparablecountablefirst-countablesecond-countableLindelöfσ-compactcountablycompactsubspacesconnectedclopen setslocally connectedtotally disconnectedpath-connectedlocally path-connectedarc-connectedinjectivesimply connectedhomotopiclocally simply connectedsemi-locally simply connecteduniversal covercontractibleidentity maphyperconnectedultraconnectedindiscretetrivial topologyopen coversequentially compactcountably compactpseudocompactparacompactlocally compactmetrizablemetric spacePolishBaire spacemeagredoor spacehomogeneoustopological groupsAlexandrovfinitely generatedcategory of topological spaceszero-dimensionalinductive dimensionBooleanStone spacesBoolean algebrasReidemeister torsionaccumulation pointmetric spacescompletenessCharacteristic classCharacteristic numbersChern classEuler characteristicFixed-point propertyHomologycohomologyHomotopy groupCohomotopy groupKnot invariantLinking numberList of topologiesQuantum invariantTopological quantum numberWinding numberIsrael Journal of MathematicsMunkres, James R.Prentice-Hall