[b]证明[/b].设[tex=2.0x1.357]khGQOVqy3eZik4Tp7/+YjA==[/tex]上的全体连续函数所作成的集合为[tex=3.5x1.357]xLOILRclS6+JKue/M0pTgw==[/tex]，则设[tex=7.071x1.357]10FzSsU3qu5IHnH/nSQjFDTOf+mqjHz8CIwVNdqeTz+np5zbkiip/07koKaLdIW6[/tex]于是[tex=8.571x1.357]JlYxpHSHqWQfLCOA9+5WuE5B6iKoCXH9HZry6chwKVzatfZ928AICG8R0Ke1ygKJeYi5zsOcTu+PRVg4sgADaw==[/tex]是[tex=2.0x1.357]khGQOVqy3eZik4Tp7/+YjA==[/tex]到[tex=3.5x1.357]xLOILRclS6+JKue/M0pTgw==[/tex]的单映射，[tex=5.286x1.786]ZVq+nZtYe1aPEQysmZ94wijC5XSanrxSeyfTWqHzVanFrS1dFySx38+unF/fT4qvg6JBowQ+/ZfNjM7cFD8t2A==[/tex].另一方面，任取[tex=4.857x1.357]Sg85sqAJ+Am0b93ti5P+4lIZ1Q1s1oiue2eUKKJ8SfA=[/tex].定义其到[tex=1.214x1.429]CkRKSvQCh8246VsHrp+7Jajeb8RpKFcbYWscBBHyDk4=[/tex]的幂集的一个映射[tex=0.5x1.214]++gGGJQcubEKWlse37f6tQ==[/tex]为[tex=13.429x1.357]5AoQzquQACNebTdNxV6/A5VzEBu6KvQ9mWmt50ofRyNc8+zT1FHMd5IMIt9DabFP01YqSFrZSyVZ+kD9EMHLWA==[/tex]下证其为单射.事实上若[tex=3.357x1.214]kgjEtdXNvoAk6xPDMbEOBGXFCUiWSt1L0WsR20NbZbwjXBNbwPNParHWSY8Gz2dP[/tex]，有[tex=11.143x1.357]5cv5f3EtVVxdkJx/MGuFB4p3BEQRylprjQPtL6NxZ0uUy/5PX/BspYCD2c4eCLyhwQpnDf3GsC52haKLJJCwUDerjrRJZVbpjH0voJxjSGXgeIwOlHjtgCHfoAeJE1vH[/tex]，不妨设[tex=7.143x1.357]eSQoFEmFXNwVsQv2Un4MSnoo79UuQVOs82ClUtXztJx3bJs6mXKJETPO7esqv9NTsWqsIjwiHJeUAoLpdTo5qwkWNJqAqLSlsrFROhXrxys=[/tex].由[tex=2.5x1.0]eo8fUHNBwmNgichz+9ng0Xyi5Ky2o/NSsdOt3B+RknE=[/tex]连续，有[tex=2.286x1.214]+JOkcjyRg/A85NiKwk1C3Qvt8Y+cyNCXkV2fVKMm+bk=[/tex]使[tex=6.857x1.357]eSQoFEmFXNwVsQv2Un4MSp6xLF0Hg0M28WhVT4bcrz0M2rw9+Gl/nhkQgUAHQWepfMkpKxWclzJYS327/uEosX/BRw69dVoHjREbVciCReI=[/tex].于是有[tex=2.286x1.214]op1eiv6S0bFIEkwoU6lR49y4gT2vlmv9DAdABkVW26I=[/tex]，使[tex=9.0x1.357]eSQoFEmFXNwVsQv2Un4MSp6xLF0Hg0M28WhVT4bcrz32Onn3ezIflhOFjFg1EkOtSUzP7RddwsDjXJgknC26PL5RWT9ft/H4lv20zTcItTA=[/tex]。则[tex=14.357x1.357]bRGe2q4oQ6fLE/FQgBPzz+b1KkxJe+cj5ZtbHVy1igImT2l96XJVvGbMAv65TfkKmYGmrnGdkRALNLcaaLbn7mFE23jQm5awA1LtGnuq9TXmadGrvApVKLm/aLH5fj0RKjMzAfXXpPf3YwXC8yJQVSLHk05Ikw71vnc6PFwiCCU=[/tex]于是[tex=7.357x1.357]ZWBfKQkoIxrR+Hr7to/RexNSStcwgx0o8Me54R4Ai7/jc4yilh9K8f53eirw5hf/pMGQUvVNAnjc//S8dqD4oFhDwfUBnCom8S+0S6T7+Qk=[/tex]为单射.于是[tex=7.0x1.786]Cy7qN50R1Q8xXzfyRxwY1+6qOlMKqKSWsx2TjMdQBBpy8hPEbvIhjjPAgCGVRMawkDYqWgxE3Q8gPytWp0bdRw==[/tex].又设[tex=0.643x1.0]YLjCNu3b8a8IkTrD4ZcqaA==[/tex]为[tex=2.0x1.357]khGQOVqy3eZik4Tp7/+YjA==[/tex]上的左连续的单调函数的全体所构成的集合.则令[tex=8.786x1.357]NnUhdvRyR1ELHV9EXxPTPElM+RpL+X0iKhZ+SLBvHOmbvRLIsSS1OJtmGqyH4RFF[/tex]于是[tex=2.857x1.0]hnxU3hW3Ad28FDmXJ7acAd8poZsq+IRvfNCBD3Ks7CA=[/tex]构成[tex=2.0x1.357]khGQOVqy3eZik4Tp7/+YjA==[/tex]到[tex=0.643x1.0]YLjCNu3b8a8IkTrD4ZcqaA==[/tex]的单射.[tex=2.429x1.571]ZVq+nZtYe1aPEQysmZ94wijC5XSanrxSeyfTWqHzVakpFPX28eC6U+11yUPgXW/R[/tex].令一方面.令[tex=1.0x1.214]onCpuCJ61nt47RoP/BQ8nA==[/tex]表示[tex=0.643x1.0]YLjCNu3b8a8IkTrD4ZcqaA==[/tex]中的单增函数.任取[tex=7.357x1.214]/iIQWwPuBlKNsaLreOieICZ5KsmwQBqlzgfIBRw5OrFR1ZHQtqC6HR7ZjfG9/Yk/ac3j5HcmTzN343bnl48w1g==[/tex].有[tex=3.643x1.357]5cv5f3EtVVxdkJx/MGuFB50Ybb3YhDXp8NmF5uN28ks=[/tex]，使[tex=6.857x1.357]i/EjjsdlWtEp7njjFez19mBQwfYpx0lzQu/eTZEYPHrSjvP7oDTfZp0yCz+etMY4BVAXtuqEjeafPDKG/1bT2v2u9JcE570LTsGMbnZ9aEM=[/tex].不妨设[tex=6.857x1.357]i/EjjsdlWtEp7njjFez19pQHg21IIgQU5z2NpMYSTtC8uyxpTOObTCVy0qxpnkphc0wZS57dfZcYJeCt+BB9Yg==[/tex].由左连续，有[tex=5.786x1.214]+JOkcjyRg/A85NiKwk1C3URzPLBiUkI8njm5EJde9fJn7IGnaLOJ2KzaCmC8n04I[/tex],使[tex=14.714x1.357]i/EjjsdlWtEp7njjFez19oedK9Zo7kqFpw2Aadw1gRkM8wnOT6TFbRVBQFSMHEsIcmSR7Jp8pZ8cXgt1KoiypNyLdSLpXBA5mX/oLDeo1lvVCRZ/f32kmXxzAWwgXkxHAuLHoEDJr8zl7cNW5fqoxfextqhoFAQILDgBhp9d1/g=[/tex]于是有[tex=11.429x1.357]op1eiv6S0bFIEkwoU6lR4wMJrUVcCNz3wm9acz6EHK+ja8U6J76QFfYCPc13kMCe0I/CPTiGTES0NfVHeT6UraIxOBHwhOY8ldAUxyFUW0Wp0jBmKR+oXpKMgoMlSn2q[/tex].同上，可得[tex=4.429x1.643]Cy7qN50R1Q8xXzfyRxwY18KNHfp7h7F/3042mDZAURrD+B6dhsD/FNH965pLGXWcZo/0n3b67qyg4OdETDflHQ==[/tex].又设[tex=1.0x1.214]CQQKmEkazyFAZIs70jkMWw==[/tex]为[tex=0.643x1.0]YLjCNu3b8a8IkTrD4ZcqaA==[/tex]中单减函数.[tex=6.071x1.357]wL4c2lmwgJfSmjSz1rWeAdvC+xT1DK5Z1xtINFPnOng=[/tex]为[tex=1.0x1.214]CQQKmEkazyFAZIs70jkMWw==[/tex]到[tex=1.0x1.214]onCpuCJ61nt47RoP/BQ8nA==[/tex]的一一对应.[tex=2.714x1.643]Cy7qN50R1Q8xXzfyRxwY14KwwBqUzxbDvdePh/0XSyspCbVSxjqtp4OnGd3Cpjc8wSudaR9xCWGqoFTK+QvwBA==[/tex].于是[tex=6.214x1.643]Cy7qN50R1Q8xXzfyRxwY14/seKmqO+J3BKbf6T3u7zN5tBtlquHvJD+7TJ2DxJu29rf9CxIJ2GVZi2PJsBnKXUp9gVkZGAx2nHcNFwe5tH/Pr5JosJNIVdhn8TV641bN[/tex]