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

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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

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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 | 2 Comments

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

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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

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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

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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

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