what is a quotient space?

Question

I am reading the session of quotient space in Topology by K. Jänich. He suggests a mental image for quotient spaces. But the book doesn't have explanations to this picture. For the graph on the left, I think the line at the bottom is an equivalence class $[x]$ obtained by the quotient map $\pi$. Why does he put $X/$~ on the right of this line? What's the meaning of the other line at the right bottom? Why is $\pi^{-1}(U)$ open?

Answer 1

The quotient space $X/\sim$ comes equipped with a canonical function $\pi:X\to X/\sim$ which sends each point $x\in X$ to its equivalence class $[x]\in X/\sim$. Until you have specified a topology on $X/\sim$, this is only function (mapping between sets) not a continuous function between spaces.

The definition specifies what to take as open sets: those sets $U\subset X/\sim$ such that $\pi^{-1}(U)$ is open. This is the coarsest topology which makes the function $\pi$ continuous.

The picture on the left illustrates what the function $\pi$ does. All the points on the squiggly line are representatives of the equivalence class $[x]$. $\pi$ maps each point to the single point $[x]\in X/\sim$. The line at the bottom on the left is the space $X/\sim$.

The picture on the right illustrates what the preimage of an open set in $U\subset X/\sim$ under the map $\pi:X\to X/\sim$ looks like. The line at the bottom right again represents the space $X/\sim$.

Answer 2

The picture is meant to be suggestive more than anything else. You can think of the picture on the left as saying "since every element on the vertical squiggle is sent to the same point under $\pi$, we say that they all belong to $[x]$."

Furthermore, the point is to illustrate that there is an identification between partitioning your set $X$ by some equivalence relation and considering equivalence classes, and also choosing representatives from each equivalence class and calling this new space $X/ \sim$.

The picture on the right, is showing how we endow this set with a topology. If $\pi^{-1}(U)$ is open in $X/\sim$, then we declare $U$ to be open. That's the definition. A different way of thinking about this, is that it is the coarsest topology that makes the quotient map continuous.