I have a very naive question about the product topology. Whenever I see a topological space with a product topology structure, this structure is already explicitly given. That is, the statement will begin something like "$X\times Y$ be a topological space...". My question is somehow "(how) can you know a topological space has a product structure?". If you are given a topological space X, are there some necessary and/or sufficient conditions which allow you to test whether there is a non-trivial Y and Z such that $X\cong Y\times Z$?
My naive guess would go something like: Let $X$ be a topological space (maybe with some special structure). Let $\sim$ be an equivalence relation (maybe with some special structure) and $X/\sim$ be the equivalence classes equipped with the quotient topology. Now consider each $[x]\in X/\sim$ as a subset of $X$, and equip each of theses subsets with the subset topology. Then, if $[x]$ is homeomorphic to some topological space $Y$, $\forall [x]\in X/\sim$, then $X \cong X/\sim \times Y$?
I'd guess this isn't true, but I haven't been able to come up with (what i am sure is) an obvious counter example. I believe my intuition about product spaces vs quotient spaces is deeply flawed, so some thoughts on this would also be appreciated.
Let $\mathcal{M}$ be the Mobius strip and let $p\colon\mathcal{M}\to S^1$ be the map that retracts the Mobius strip in its central circle. I define the equivalence relation on $\mathcal{M}$: $$x\sim y \iff p(x)=p(y).$$
Can you see that this is a counterexample?