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[美] 默可雷斯 著 / 机械工业出版社 / 2004-02 / 平装
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拓扑学:第2版
本书分为两个独立的部分。第一部分普通拓扑学,讲述点集拓扑学的内容:前4章作为拓扑学的引论,介绍作为核心题材的集合论、拓扑空间、连通性、紧性以及可数性和分离性分理;后4章是补充题材。第二部分代数拓扑学,讲述与拓扑学核心题材相关的主题,其中包括基本群和覆盖安闪及其应用。
本书最大的特点在于对理论的清晰阐述和严谨证明,力求证读者能够充分理解。对于疑难的推理证明,将其分解为简化的步骤,不给读者留下疑惑。此外,书中还提供了大量练习,可以巩固深学习的效果。严格的论证、清晰的条理、丰富的实例,让深奥的拓扑学变得轻松易学。
refaceanotetothereaderpartigeneraltopologychapter1settheoryandlogic1fundamentalconcepts2functions3relations4theintegersandtherealnumbers5cartesianproducts6finitesets7countableanduncountablesets8theprincipleofrecursivedefinition9infinitesetsandtheaxiomofchoice10well-orderedsets11themaximumprinciplesupplementaryexercises:well-orderingchapter2topologicalspacesandcontinuousfunctions12topologicalspaces13basisforatopology14theordertopology.15theproducttopologyonxxy16thesubspacetopology17closedsetsandlimitpoints18continuousfunctions19theproducttopology20themetrictopology21themetrictopology(continued)22thequotienttopologysupplementaryexercises:topologicalgroupschapter3connectednessandcompactness23connectedspaces24connectedsubspacesoftherealline*25componentsandlocalconnectedness26compactspaces27compactsubspacesoftherealline28limitpointcompactness29localcompactnesssupplementaryexercises:netschapter4countabilityandseparationaxioms30thecountabilityaxioms31theseparationaxioms32normalspaces33theurysohnlemma34theurysohnmetrizationtheorem35thetietzeextensiontheorem36imbeddingsofmanifoldssupplementaryexercises:reviewofthebasicschapter5thetychonofftheorem37thetychonofftheorem38thestone-cechcompactificationchapter6metrizationtheoremsandparacompactness39localfiniteness40thenagata-smirnovmetrizationtheorem41paracompactness42thesmirnovmetrizationtheoremchapter7completemetricspacesandfunctionspaces43completemetricspaces44aspace-fillingcurve45compactnessinmetricspaces46pointwiseandcompactconvergence47ascoli'stheoremchapter8bairespacesanddimensiontheory48bairespaces49anowhere-differentiablefunction50introductiontodimensiontheorysupplementaryexercises:locallyeuclideanspacespartiialgebraictopologychapter9thefundamentalgroup51homotopyofpaths52thefundamentalgroup53coveringspaces54thefundamentalgroupofthecircle55retractionsandfixedpoints56thefundamentaltheoremofalgebra57theborsuk-ulamtheorem58deformationretractsandhomotopytype59thefundamentalgroupofsn60fundamentalgroupsofsomesurfaceschapter10separationtheoremsintheplane61thejordanseparationtheorem62invarianceofdomain63thejordancurvetheorem64imbeddinggraphsintheplane65thewindingnumberofasimpleclosedcurve66thecauchyintegralformulachapter11theseifert-vankampentheorem67directsumsofabeliangroups68freeproductsofgroups69freegroups70theseifert-vankampentheorem71thefundamentalgroupofawedgeofcircles72adjoiningatwo-cell73thefundamentalgroupsofthetorusandtheduncecapchapter12classificationofsurfaces74fundamentalgroupsofsurfaces75homologyofsurfaces76cuttingandpasting77theclassificationtheorem78constructingcompactsurfaceschapter13classificationofcoveringspaces79equivalenceofcoveringspaces80theuniversalcoveringspace81coveringtransformations82existenceofcoveringspacessupplementaryexercises:topologicalpropertiesandchapter14applicationstogrouptheory83coveringspacesofagraph84thefundamentalgroupofagraph85subgroupsoffreegroupsbibliographyindex
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