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The following is a theorem in topology:

Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous map such that $a \sim b$ implies $g(a) = g(b)$ for all $a$ and $b$ in $X$, then there exists a unique continuous map $f : X/\sim → Z$ such that $g = f ∘ \pi$.

In Wikipedia, it is said that the quotient space $X/\sim$ together with the quotient map $\pi : X → X/\sim$ is characterized by the the theorem above, which is also called a universal property.

I know almost nothing about category theory. The Wikipedia article about universal property says that

  1. The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construct is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details.
  2. Universal properties define objects uniquely up to a unique isomorphism. Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property.

Here are my questions:

  • What does "$X/\sim$ together with $\pi$ is characterized by the theorem" mean?

  • It seems to me that the "universal property" can be used to define something. Does this suggest that the theorem above can be used to define the quotient topology and the projection map?

  • Could anybody give me a simple example about how the theorem above can be used to make some proof shorter (and how complicated a proof could be without using it)?

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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.

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  • $\begingroup$ Hausdorffness is not necessary for the theorem (and the first sentence of its proof) to be true :-) $\endgroup$ – Stefan Hamcke Mar 14 '16 at 14:08
  • $\begingroup$ Finally a great example of how to apply category theory in real action. I guess this exactly uses the adjoint function theorem? Thanks a lot!!! (+1) $\endgroup$ – C-Star-W-Star Jun 15 '16 at 10:45
  • $\begingroup$ I guess this kind of strategy also applies to the problem: Characterize all ideals of a quotient algebra! $\endgroup$ – C-Star-W-Star Jun 15 '16 at 10:47

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