general topology - muller

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1、GENERALTOPOLOGYJESPERM.MØLLERContents1.Sets,functionsandrelations31.1.Sets31.2.Functions31.5.Relations42.Theintegersandtherealnumbers73.Productsandcoproducts94.Finiteandinfinitesets105.Countableanduncountablesets126.Well-orderedsets147.Partiallyorderedsets,TheMaximumPrincipleandZorn’slemma168.Topolo

2、gicalspaces188.4.Subbasisandbasisforatopology189.Ordertopologies2010.Theproducttopology2010.4.Productsoflinearlyorderedspaces2110.6.Thecoproducttopology2111.Thesubspacetopology2111.5.Subspacesoflinearlyorderedspaces2212.Closedsetsandlimitpoints2412.3.Closureandinterior2412.10.Limitpointsandisolated

3、points2512.14.Convergence,theHausdorffproperty,andtheT1-axiom2613.Continuousfunctions2713.6.Homeomorphismsandembeddings2813.13.Mapsintoproducts2913.17.Mapsoutofcoproducts3014.Thequotienttopology3114.1.Openandclosedmaps3114.2.Quotienttopologiesandquotientmaps3115.Metrictopologies3715.6.Thefirstcountab

4、ilityaxiom3715.13.Theuniformmetric3916.Connectedspaces4017.Connectedsubsetsoflinearlyorderedspaces4217.5.Pathconnectedspaces4317.9.Componentsandpathcomponents4317.12.Locallyconnectedandlocallypathconnectedspaces4418.Compactspaces4719.Compactsubspacesoflinearlyorderedspaces5019.6.Compactnessinmetric

5、spaces5120.Limitpointcompactnessandsequentialcompactness5321.LocallycompactspacesandtheAlexandroffcompactification5322.Countabilityaxioms57Date:2ndApril2005.12J.M.MØLLER23.SeparationAxioms5824.Normalspaces6025.SecondcountableregularspacesandtheUrysohnmetrizationtheorem6225.1.Anembeddingtheorem6225.5.

6、Auniversalsecondcountableregularspace6326.CompletelyregularspacesandtheStone–Cechcompactificationˇ6526.6.TheStone–Cechconstructionˇ6527.Manifolds6828.Relationsbetweentopologicalspaces69References70GENERALTOPOLOGY31.Sets,functionsandrelations1.1.Sets.Asetisacollectionofmathematicalobjects.Wewritea∈Si

7、fthesetScontainstheobjecta.1.1.Example.Thenaturalnumbers1,2,3,...canbecollectedtoformthesetZ+={1,2,3,...}.Thisna¨ıveformofsettheoryunfortunatelyleadstoparadoxes.Russel’sparadox1concernstheformulaS6∈S.Firstn