I hope someone will read this long question to the end, I'm getting really desperate.
Here's what I have to prove:
Let be defined as follows:
where B is the Polish set with product topology and every has discrete topology.
Show that is a homeomorphism.
Injectivity and surjectivity is easy, continuity isn't.
Here are the facts:
is continuous if and such that .
is continuous if and only if is continuous, for all i.
Here's what I got, after huge online help:
Let be the inverse.
is continuous if and are continuous, where
takes to .
, akes to .
Since and are also products, we can say that is continuous if , and , are continuous.
sends to the i-th coordinate of the part, and sends to the j-th coordinate of the part.
Now it's easy to prove that the inverse is continuous.
The question is: what do I do about ?