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Monthly Archives: May 2011
Product of sequence of Polish spaces is Polish
The product of a sequence of Polish spaces is Polish. A Polish space is a topological space that is separable and completely metrizable. Let be a sequence of Polish spaces. Let with product topology . Then, since each is completely … Continue reading
Posted in Descriptive Set Theory, Kechris
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Product of sequence of separable spaces is separable (p3 (A))
The product of a sequence of separable topological spaces is separable. Proof. Let be a sequence of separable spaces. For each , let denote a countable dense subset. Let have the product topology, . Let , . Let , which … Continue reading
Posted in Descriptive Set Theory, Kechris
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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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Lemma (p13 (D))
Let be a metric space. Then is a compatible metric. Furthermore, is complete implies is complete. Proof. Let . Let . Now implies . So . Also, if , implies , so . Now, assume that is complete. Then let … Continue reading
Posted in Descriptive Set Theory, Kechris
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Convexity
Rudin RC p. 61 Let , . is called convex if holds whenever , . Show that this is equivalent to whenever . It is easy to see that is equivalent to , , such that , and . Now, … Continue reading
Posted in Analysis, Rudin RC
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Hello world!
I have decided to start writing a blog of math. I’m not sure exactly what I will post here yet. Most likely, every so often I will post a proof that I think is interesting. Also, I might attempt to … Continue reading
Posted in Uncategorized
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