# 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

## 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 | 1 Comment

## 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

## 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