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Category Archives: Logic
The unit ball of Hilbert space is not compact (p18 (B))
Show that the unit ball of Hilbert space is not compact. Proof. Let . For each , define by , if , and . Let . Now, let . Then , where , and . Now, for each , for … Continue reading
Posted in Descriptive Set Theory, Kechris, Logic
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Product of a sequence of completely metrizable spaces is completely metrizable (p13 (C))
The product of a sequence of completely metrizable spaces is completely metrizable. Proof. Let be a sequence of completely metrizable spaces. For each , let denote a compatible metric. Let have the product topology, which is given by the metric … Continue reading
Posted in Descriptive Set Theory, Kechris, Logic
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Continuous distance (p13 (E))
Let be a metric space. Let . Let by . Then is continous. Proof. Let . Let be a basic neighborhood of in . Then , for some . Let , . Then . Also, , so . Therefore, is … Continue reading
Posted in Descriptive Set Theory, Kechris, Logic
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