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.

  • $\begingroup$ $X = X \times \prod_\mathbb{R} *$? $\endgroup$ – Najib Idrissi Jul 7 '14 at 8:00

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.

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