Intuition behind the notion of quotient topology.

Question

Let $X$ be a topological space and $f: X\to Y=X/\sim$ be the map $x\mapsto [x]$.In order to make $f$ continuous,$Y$ needs to be given a topology $\{U\subset Y: f^{-1}(U)$ is open in $X\}$.This topology is called the quotient topology. I want to know the motivation behind this definition. I also want to know why it is the largest topology on $Y$ that makes $f$ continuous.

Answer 1

I'll answer your second question first. Suppose we have a topology $\mathcal{T}$ on $Y$ which is strictly larger than $\{U\subset Y: f^{-1}(U) \text{ is open in X }\}$. Well then necessarily we must have some $V\in \mathcal{T}$ such that $f^{-1}(V)$ is not open in $X$, so $f$ cannot be continuous.

To motivate why we might want this topology consider a map $\varphi:X/\sim=Y\to Z$ where $Z$ is some other topological space. Then if $Y$ has the quotient topology the map $\varphi$ is continuous if and only if $\varphi\circ f:X\to Z$ is continuous. Moreover, the quotient topology is the only topology on $Y$ for which this is true. The quotient topology also gives the nice property that if we have a continuous map from $X$, say $\phi:X\to Z$, that respects the quotient (i.e. $\phi(x)=\phi(y)$ whenever $f(x)=f(y)$ ) then $\phi$ descends uniquely to a continuous map $\tilde{\phi}:Y\to Z$ with the property that $\phi=\tilde{\phi}\circ f$.

For proofs of my claims or perhaps more motivation I would recommend you look at the section on Quotient spaces in Lee's Introduction to Topological Manifolds.