• 2022-05-29
    证明每一坐标空间都同胚于积空间的某一子空间。
  • 证:设[tex=4.429x2.857]fGxAv5BnoECiuaBzLGidJTnsiDYH1pkXZuyrcw/YNLHS/76hxj3BMtPQM5h5qOiv[/tex]是拓扑空间族[tex=3.5x1.571]Z83fl0p/j+tpzWcjS+ikCISvAjqndFHpiD1t25KpE16MZ34q/uT8x1s6woUBTExY[/tex]的积空间,取定[tex=2.429x1.214]dO4DRvySlHyZkwI0j+Ry9g==[/tex],对任一[tex=5.571x1.571]g+jrmGFonhdbumNAvMih38R8J4gKUeWUpFGBiUEiKB8OqDsaGx9z1YS2fyGaQA3nIusBcw6n8bmOaQCanrwUnQ==[/tex],定义[tex=17.429x3.429]9ZzMk2m+15jXYvjeAP0KGaQS+5Z1vukllxoB+s5SdeLfJBhGfvuTTuj2m4fIWH2VzfimRLNvl7W/YCNpUE2enYaa27GCjV9iSv2LlNGQiNG4ZoWlSGloFd6XC/c83w5X2z1Q5RtinCpYQO5slRMDxVnF7fTq7ZFclB0IOsKBFxfXGYf5qO+6uIAi/HyYITt898AXHjUq0v6zRXsA+3yl9hoP7apnaRT8+hyoZTQBys9SAhZohYqFzfAO8TnFS/pM[/tex]以及[tex=3.786x1.357]3yJErkN8OyBbnCckEVBien5/V3j8muz5z2FDkF3Kq+Q=[/tex]易见,[tex=4.143x1.286]lrziNDLDFK8+BP4NRAb3eGVGHFwfoZ5eme4Wm2tLfjY=[/tex]是一一在上的连续映射,下证[tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex]为开映射。对[tex=0.857x1.286]h9C4nePGcGllh55hxKIsUw==[/tex]的基[tex=0.929x1.0]JO9Qe4aXLr5u0mzxwsuhiw==[/tex][tex=35.214x1.357]w76mAH9J+ANoHlNaETQyqnR2GYTo4qdheElvV6+qBPsZ0VSaoUSVunAksYQmW9XHOaK2EM3VsFVqSH+aqsGFZ17t84Qupa8Pj3EGuepdc9IsNfQ/ryIQY57+vrPUY1zQWntTV7ajLKEIQQn37OAXCIwRnPAhl4Z4NASNZveI0DPHZ5ryPONPcaZ3s2xkCVmgUCMLHT0dJSFdfnKCf93+tOpeN9sx7YbAuzveeFAraoFLgATHHvzYo+PcY5FABxcnfIcW8Ui9KkHaQBCbs2lK16ru0EKLK87XkDvQnmBwifyQG5n6DBf3MkKQSisrqa+HHC5fqTZ8Y8JXP5VKyQM9avTMMv6ii06KdXOPNuqAbHg=[/tex]中的任一非空成员[tex=19.0x1.643]jc8hflHaW03YGcNX7YreroQCU1X9L9mGGDVbrV1Wq45+igWIQGziLAHm/3P86p9kjL64in288ajoxprR4JIJEVfU+f3TexC/lMu/hw8etx1bfglOgXQ+JEUCAQ/WTAcsl4eWCTD+LrQVuJy8quw9fBpPLZFwfA6WSkop2MbNZIpzbXkJVJQHSkzcsTarVYZTL1dFOaY2xhfBlFmp35X00lbv6rYNqvZXejfMYR9Dg3VC6Pg2WiuH66XSRaKiJl69[/tex]不妨设[tex=5.143x1.0]VOIOg6+tdx2UICroZLOhnOly3y9BFtn66+Y2+ILOCTDZWwKr44Pey+MsQviPOwGS[/tex]互不相等,且[tex=8.0x1.357]OPG1tE4fbAinCDnGLhqNi4z1YwqIDBM0631tNCl8rXInQ6dnc6z7V4LCAoSck3gnK0Ax3UR7VgBYx/CirMqqrBue6H2gj5geTUPbsZAt4Do=[/tex](必要时可置[tex=3.143x1.286]RP4//0fXrOdCY68xgyCO79J861SWJHfA1dk9wp7SN+s=[/tex]),又[tex=4.143x2.857]HoWMv9Dmhxkq9SiE/hPZcIeieMp3w80Oqj2mwCw3aQ2Icp0srvmbmwAiPxf3Utu4[/tex],其中[tex=27.643x3.357]ZKJZOXsFyvvv39SaUfoCYIK48k+CJUPvsGTFpUMcZbqfaLIf01EnT57MoX3NMBJ1dqDZQlr1IdACTJua8PswzAigSlo/lOFDnYjQPIYke8ZA/Gk8WAbnSvpZRgwfXdOue2j1MxhLSSCdPFFs31+AR55UknRy8BJRLUukh+W9XOzyIx5urhsedrPvEH0qgvu+58iC8JQi2Zjt0TAjTNQVABxeYoqmqB2NPEpdVw29SfcKB/3mFg7nEQEJAirKY54radReUHhmnxYtJOl2KzUZJhce1NoH2AuoBz6mzN4D5hiA+NXHVeJYg69GtDrDmW43bTqk6pyk1aJDk/otjcdFP2N+HO4Mk8ccJRcyxLupYlM81AZqZ5EPgyezFBzu0tCDnap+JDXrC7+m40CIyPGIouBisMeAb3iKf9r/JePQSsA+lHkq9N6KDvHc3Pqc4cDz2/tZ72z9NrT5f9/jAF/nuQymSxeCmCJ6C6VhN20RZ2I=[/tex]于是[tex=9.5x3.429]Ju8Mh4j5Lu6hBT6+wvWI6P/kg/mpDkwvXIIvPry+aAJlI9RUdnZbZoLxwQV4RLbz2WHgPAoocJdg4/4i7/iHuQIhGqJOIh8hADHQSPj2ADQ=[/tex]是开集,从而[tex=0.643x1.286]+RQz+inOZSqc5WvKyEpD0Q==[/tex]是开映射。综上,[tex=1.286x1.286]pgVVu+2CRcRWnH8lbWUfBQ==[/tex]同胚于[tex=0.857x1.0]KGogyvwDAIJf/iL0H/9wjg==[/tex]的子空间[tex=0.786x1.0]XGN/WPVR/dY1VDbqK+D5fA==[/tex]。

    内容

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