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 theinverse.

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 ?

Please advise.

Thank you!