proof of Baire space is universal for Polish spaces

Let X be a nonempty Polish spaceMathworldPlanetmath. We construct a continuousMathworldPlanetmathPlanetmath onto map f:𝒩X, where 𝒩 is Baire spacePlanetmathPlanetmath.

Let d be a complete metric on X. We choose nonempty closed subsets C(n1,,nk)X for all integers k0 and n1,,nk satisfying the following.

  1. 1.


  2. 2.

    C(n1,,nk) has diameter no more than 2-k.

  3. 3.

    For any k0 and n1,nk then

    C(n1,,nk)=m=1C(n1,,nk,m). (1)

This can be done by inductionMathworldPlanetmath. Suppose that the set S=C(n1,,nk) has already been chosen for some k0. As X is separablePlanetmathPlanetmath, S can be covered by a sequence of closed sets S1,S2,. Replacing Sj by SjS, we suppose that SjS. Then, remove any empty setsMathworldPlanetmath from the sequence. If the resulting sequence is finite, then it can be extended to an infiniteMathworldPlanetmath sequence by repeating the last term. We can then set C(n1,,nk,nk+1)=Snk+1.

We now define the function f:𝒩X. For any n choose a sequence xkC(n1,,nk). Since this has diameter no more than 2-k it follows that d(xj,xk)2-k for jk. So, the sequence is Cauchy ( and has a limit x. As the sets C(n1,,nk) are closed, they contain x and,

k=1C(n1,,nk). (2)

In fact, this has diameter zero, and must contain a single element, which we define to be f(n).

This defines the function f:𝒩X. We show that it is continuous. If m,n𝒩 satisfy mj=nj for jk then f(m),f(n) are in C(m1,,mk) which, having diameter no more than 2-k, gives d(f(m),f(n))2-k. So, f is indeed continuous.

Finally, choose any xX. Then xC() and equation (1) allows us to choose n1,n2, such that xC(n1,,nk) for all k0. If n=(n1,n2,) then x and f(n) are both in the set in equation (2) which, since it is a singleton, gives f(n)=x. Hence, f is onto.

Title proof of Baire space is universal for Polish spaces
Canonical name ProofOfBaireSpaceIsUniversalForPolishSpaces
Date of creation 2013-03-22 18:46:57
Last modified on 2013-03-22 18:46:57
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 54E50