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
- 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.
- 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)?