Let $E$ be some set and $\varphi,\eta$ two functions with codomain $E$. Consider the equivalence relation generated by $\varphi(x)\sim\eta(x)$. Do we ever care about equivalence classes $X$ of $\sim$ which are not of the form $[\varphi(x)]$ (or $[\eta(x)])$?
For example, in Algebra when we construct the tensor product of two modules we only care about the equivalence classes of the form $x\otimes y$; the rest are just noise. Is the situation similar in Topology? I ask this because in topology one usually finds equivalence relations that are generated by some other relation, and I always find it difficult to workout the form of those equivalence classes which don't contain the generators (e.g. take any gluing construction you can find in a algebraic topology book).
I would feel much better if I could ignore–without any repercussions–the equivalence classes not containing the generators.
The truth is that forming "generated" quotients in Algebra is a lot simpler than in Topology: we only have to quotient by the subobject generated by those relations we require to hold in the quotient object. On the other hand, in Topology we have to do with the cumbersome connected-components construction ($x\sim y$ if there is a finite sequence $a_0...a_n$ such that $a_0=x$ and $y=a_n$ such that etc. etc.) which doesn't really help you see what the equivalence classes are like.
TD;LR: How to become more comfortable with generated equivalence relations used in (algebraic) topology?