Can any "relevant" topological spaces be decomposed into an uncountable product?

Question

Can any "relevant", as meaning generally useful topological spaces be decomposed into an uncountable product of other topological spaces with the product topology?

Many thanks in advance.

Answer 1

That is the case for some compactness arguments.

For example, have a look at the proof of the Banach-Alaoglou theorem (http://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem#Proof).

Here, one considers the unit ball in $X'$ as a subset of

$$ \prod_{x \in X} \overline{B}_{\Vert x \Vert}(0), $$

where the ball is formed in $\Bbb{K}$ (i.e. $\Bbb{R}$ or $\Bbb{C}$).

One then uses Tychonoff's Theorem to conclude the compactness.