Understanding / learning how to work with quotient spaces

When I try to work with quotient spaces in topology, I find myself stuck and confused.
I always find that the definition of a quotient space: $$ \tau_{X / \sim} = \{ V \subseteq X /\sim \, \mid q^{-1}(V) \ \text{is open on} \, X \}$$ Is not really comfortable to use. The definition and the map look so "clumsy", I feel like it's difficult to find open or closed sets in such a space.
I also had the same problem with product space, but I eventually figured it out by understanding how neighborhoods behave in such spaces. However, in quotient spaces, we don't even have a basis to work with!
I think I am missing something crucial with how to work with such spaces. If anybody has a source, or perhaps tips/tricks on how to work with quotient spaces, It'll really help.
Thanks to all who respond!

Solution 1:

I think most algebraic topologists would agree that such definitions are clumsy and not fun to work with, which is precisely why the categorical point of view is lauded. Quotients $X/\sim$ are characterized by a certain universal property: if $g:X \to Y$ is a continuous map such that $a \sim b$ implies $g(a)=g(b)$, then $g$ uniquely factors through the quotient map $q:X \to X/ \sim$.

In many cases (but perhaps not all) this universal property suffices to prove most things we need to prove about quotient spaces.

If you want to build some intuition as to the set-theoretic definition you've provided, I would recommend thinking of some simple examples. Consider $[0,1]$ with the endpoints identified. Verify that the quotient topology on this quotient space is consistent with the usual topology on $S^1$.