proof of Baire space is universal for Polish spaces
has diameter no more than .
For any and then
This can be done by induction. Suppose that the set has already been chosen for some . As is separable, can be covered by a sequence of closed sets . Replacing by , we suppose that . Then, remove any empty sets from the sequence. If the resulting sequence is finite, then it can be extended to an infinite sequence by repeating the last term. We can then set .
We now define the function . For any choose a sequence . Since this has diameter no more than it follows that for . So, the sequence is Cauchy (http://planetmath.org/CauchySequence) and has a limit . As the sets are closed, they contain and,
In fact, this has diameter zero, and must contain a single element, which we define to be .
This defines the function . We show that it is continuous. If satisfy for then are in which, having diameter no more than , gives . So, is indeed continuous.
|Title||proof of Baire space is universal for Polish spaces|
|Date of creation||2013-03-22 18:46:57|
|Last modified on||2013-03-22 18:46:57|
|Last modified by||gel (22282)|