universal property in quotient topology

Solution 1:

1) "Characterized by" means that it is a complete description. A topological space $Y$ is homeomorphic to $X/\sim$ if and only if it satisfies the universal property.

2) A universal property can be used to characterize something if it exists. So you can say that an object $Y$, if it exists, is defined by the following universal property... However, showing existence is a separate issue.

3) Here is a theorem whose proof is quick and clean once you have some categorical properties established. Note that the messy details are hidden in the first line of the proof. If you write out these details, the result is worse than the point-set proof.

Theorem: Let $\sim$ be an equivalence relation on $X$ and let $Y$ be a locally compact Hausdorff space. Then $(X\times Y)/\sim\, \approx \,(X/\sim) \times Y$.

Proof: Since $Y$ is locally compact Hausdorff, the functor $-\times Y$ admits a right adjoint, hence $-\times Y$ commutes with colimits. The universal property of $X/\sim$ is a colimit construction.