Let $\tau$ be a topology on a topological space $X \times Y$ which is not a product topology. Consider the subspace topology $\tau_X$ and $\tau_Y$ induced by the topology $\tau$. My question is would the product topology $\tau_X \times \tau_Y$ on $X \times Y$ coincide with the topology $\tau$.

I tried proving this by showing open sets in one topology is contained in other. I think I can show that $\tau \subset \tau_X \times \tau_Y$ but I am not able to show the other way round.

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    $\begingroup$ $\tau$ is not a product topology and $\tau_X\times\tau_Y$ is a product topology. This is enough to conclude that the topologies do not coincide. $\endgroup$ – drhab Dec 7 '18 at 15:59

You are considering subspace topologies on what subsets? $X$ and $Y$ are not subsets of their cartesian product; you could embed them via some injection, for which on the other hand there is no canonical choice (and would not even exist in the extreme case when only one of your two sets is nonempty).


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