TITLE: Topological Version of First Isomorphism Theorem
QUESTION [5 upvotes]: Given a set $X$ and an equivalence relation $\sim$ on $X$, we can define the set $X_\sim=\left\lbrace\left[x\right]:x\in X\right\rbrace$ of equivalence classes, and we can define a projection map $\pi:X\rightarrow X_\sim$ defined by $\pi(x)=\left[x\right]$. If we now put a group structure on $X$, then there are some equivalence relations that are special: if we let $x\sim y$ iff both elements are in the same coset of a normal subgroup of $X$, we can define a group structure on $X_\sim$ in a natural way such that the map $\pi$ is a homomorphism, which is just the first isomorphism theorem.
My question is: Is there a corresponding situation in the case of quotient spaces of topological spaces? For any equivalence relation on a topological space, we can define the quotient topology on the quotient space, making the projection map $\pi$ continuous, which seems to be the sort of preservation of structure we might be after. So are there any relations that are special in some way analogous to the way that those arising from quotienting by a normal subgroup are special in the group setting? And do these arise in any way from topological spaces in the way that the ones in the group setting arise from groups?

REPLY [5 votes]: When $ S$  is a topological space, and $E$ is an equivalence relation on $S$, the natural- quotient-space topology on the set $S_{/E}$ of $E$-equivalence classes is defined to be the strongest topology on $S_{/E}$ such that the function $f_E(x)=[x]_E $ ,...for $x\in S$ ..., is continuous, (where $[x]_E$ is the $E$-equivalence class containing  $x$.  ). So  any $ V \subset S_{/E}$ is open in $ S_{/E}$ iff  $f_E^{-1} V = \{ x \in S : f_E(x) \in V \}$ is open in $ S$  .  The set $ S_{/E}$ with this topology is called the quotient space and $f_E$ is called the quotient mapping. This is a large area of study, comparable to the size and scope of  quotient groups in group theory.  In particular, when $T$ is a subset of $S$,and $E$ is defined by [  $ xEy $ iff $($  $x=y$ or  $\{x,y \} \subset T $ $)$ ],  then the quotient map $f_E : S \to S_{/E}$ is called the identification of $T$ to a point( which is useful in constructing many examples of certain properties).