证：设[tex=0.929x1.0]yrdxKP53BahSSWdMks+nxQ==[/tex]为满足第二可数性公理的拓扑空间 [tex=0.857x1.0]KGogyvwDAIJf/iL0H/9wjg==[/tex]的基，而[tex=1.286x1.286]qcE83sM7A09zrkagztDS/kqZJLPE7XTIkpwjw3MtlNI=[/tex]是[tex=0.857x1.0]KGogyvwDAIJf/iL0H/9wjg==[/tex]的一个可数基，于是对任意的[tex=2.714x1.214]1JBs/6ePNnmfADU++y9KHt/zjVVKsTg3D8EPf9bgZAQ=[/tex]，存在 [tex=3.429x1.429]uTlCkmfeVWu72lLbxxUQo+naPsQRlZHWeeoQZUKwMdUj9jDPS5qbPPUwr3I/4tKz[/tex]，使 [tex=7.643x2.286]8jlM7gyHT7328H7t1M5ZzkGdRbj6rrterNz1fA4iRIhAX0Zrf5FD92ItIFfrxx6zagtjlVVcW8h1DAMRlfDWiQ==[/tex]对任意的[tex=3.143x1.429]PlTBIFwv09pGGbZ33FqTrOgtEi8ifaY3pC8d9TohVMo=[/tex]，存在[tex=3.929x1.357]KO76kV94H0QMmJhQF2idGc6hO4eEBT/ElFiWQ2B7A2G6VOoyEKeN+UiUBPweeDF3[/tex]，使[tex=7.357x2.143]PLu2TYA5pNo5QX62Hc+jZF56wv3R65OL6WSwlRyDS1bdY/eTVJ3QDws1FBTVDPxR[/tex]令[tex=6.286x2.214]1sQvQXgp6FkBgSy3cA1jxeVaje+1Dn3jWsMkNUCnHXFrS2cOa76SXkS69GZvpjEAXszXTKqN54KqR/rv7MSYUmgfRHzpBQMXiJiVMCPkZwE=[/tex]，则[tex=2.857x1.286]Sa6d/RLDBSKLDj+2aJOqmljlXqpyvGngflqsElPu5/xuHIaWKNm50r9Zv+20/0oM[/tex]，从而[tex=1.143x1.0]ct1heifmhlRlaKf9IxeRz7R6yuvApP6hxhsdYRIkYc4=[/tex]可数且[tex=19.143x3.0]32sj2NlaUwecRKzq1ww/0uU86nejZv2+iXBCgWvViDLMrR1wxqAwoL6V1SGe6KZ3w54jyk//OwdSHqKX1IK3TsHlEKD3q6fdLTlEI8L8NSezfp7qeOn5GQOvG18rgWG48jlJNd6QiteN2zqRFQLUDQCey3EU/0/1gEEnkLa1+EFPzDV3/NhwMrZ3fQTgPE6O6DWnjlBwE8d9rzxPFEu2QA==[/tex]据(2)，对每个[tex=2.357x1.071]e0i2XE+oVw9K0tghrdtE7pHsCtMqaksdj5gi1D5CYLE=[/tex]，可选取一个[tex=3.643x1.429]Hd18m5tVTX/9zmSOpletZ8SBa6+FVS8OWOUfnXi+phw=[/tex]，使 [tex=2.786x1.214]pQN6PXJo1nczRP0ltD6w7bK2BlppYvv9rdE3D6yHLCo=[/tex]，令[tex=7.143x1.429]uTlCkmfeVWu72lLbxxUQo6oZvA4xeqv58Av7/dVoGVwcnOXYY7oJP38voRi8TvRU05w3RRchwg9yH/BB/zoMHG2IEjA1Jt0wSesZA8K+MoQ=[/tex]，则易见[tex=1.5x1.357]auOwfZoF1px7kD/SD7IIBx9N2k2E8M65+hPEPc579uI=[/tex]可数，且[tex=16.429x2.214]iSh94GpQL7jMBn75N0I16ZjlV8YpRnZI1wRCLmJowvvDiUNjGusKiQhFXodxmjBMQt4v+qe9rIqnobDqORObfQGutyGjiDdiWykDcprJFQHRZJDnz5urzZwerskiMvEERomH1H8bjnNiJgJHAVYYZ+dhgHh90qD70y5D5VuclMn2UakcSTPxkJy/wAGfB01U[/tex]，令[tex=6.786x2.071]HmKNoJT2jGlSpjIej7vnciSYqAISlp/TZk5+xn8psSThnos3VOO2oLNCQ8KuO7HxfzcL6ERIa6krcHb10SGt2TNEXLEL3MkbJhLz3KRfiBgJ1/plKZvUz5TBT1oAQH7o[/tex]，则[tex=1.286x1.143]HmKNoJT2jGlSpjIej7vncqEnnunvK1MlzI3CgN2s3nk=[/tex]是可数个可数族之并，因而可数，且显然[tex=3.0x1.143]HmKNoJT2jGlSpjIej7vncqLI9Ze2Ya436g6bXGUHfMjnj+PYHZYVzofd0F2afjP8[/tex]。所以[tex=1.286x1.143]HmKNoJT2jGlSpjIej7vncqEnnunvK1MlzI3CgN2s3nk=[/tex]是[tex=0.929x1.0]yrdxKP53BahSSWdMks+nxQ==[/tex]所包含的一个可数基。