Chapter 7 Infinite product spaces

7.2. PROOF OF BOREL ISOMORPHISM THEOREM 5
1
and noting that, since an = b x c, if x is irrational two distinct sequences will
an 1
lead to two distinct numbers. The fourth injection is a diagonal map: Write an
element x =(x1, x2,...) 2 [0, 1] N by means of a matrix am,n 2{0, 1} such that
Now consider the diagonal map
xm = Â n 1
k 1
f(x) =Â Â
k 1 m=1
am,n
, m 1. (7.13)
2n a m,m k
. (7.14)
21+···+k 2+m
It is easy to check that this map is continuous, one-to-one and the inverse is continuous.
By Lemma 7.4, the images of these maps are Borel and so all maps are Borel
measurable. By Theorem 7.3, all of these spaces are Borel isomorphic.
Recall the well-known fact—which enables the definition of dimension—that [0, 1]
and [0, 1] 2 are not homeomorphic. However, by the arguments in the above lemma,
as standard Borel spaces they are indistinguishable.
Having proved the Borel-equivalence of examples 2-4 in our list above, let us now
address the embedding of a general Polish space. Here it is convenient to work
with Hilbert cube:
Proposition 7.6 [Universality of Hilbert cube] Every Polish space is homeomorphic
to a Gd-subset of [0, 1] N .
Proof. Let (X, d) be a Polish space and let us assume without loss of generality
that d 1 (otherwise maps (X, d) ! (X, d
1+d ) by identity map). Let (xn) be a
countable dense set in X. Define the map f : X ! [0, 1] N by
x 2 X 7! f (x) = d(x, xn) n 1 2 [0, 1] N . (7.15)
As is easy to check, f is continuous and one-to-one and so we have f 1 : f (X) ! X.
We claim that f 1 is also continuous. Indeed, suppose that (zn) is a sequence such
that f (zn) ! f (z) in f (X). Let e > 0 and find n 1 such that d(xn, z) < e. Then
d(zm, z) apple d(zm, xn)+d(xn, z)
= f (zm)n + d(z, xn) !
n!• f (z)n + d(z, xn) < 2e. (7.16)
Hence zm ! z or, in explicit terms, f 1 ( f (zm)) ! f 1 ( f (z)), and so f 1 is continuous
on f (X). Thus f : X ! f (X) is a homeomorphism and so, by Lemma 7.4, f (X)
is a Gd subset of [0, 1] N .
The previous proposition gives the desired embedding of X into the Hilbert cube
and, by Lemma 7.5, a Borel isomorphism into the Cantor space. It remains to construct
an embedding of the Cantor space into a general (uncountable) Polish space.