Let $X$ be a topological space and assume that we are given an equivalence relation on $X$, denote it $\sim$. Then one can define the quotient space $X/\sim$.
On the other hand, if we denote by $[x]$ the equivalence classes of the elements in $X$, then every two classes are either disjoint or equal. I want to look just on the disjoint ones, i.e. to write $X/\sim= \coprod [x]$. Choose for any element in the union its representative and denote the set of represantatives by $Y$ ($x,y\in Y$ iff $[x]\cap [y]=\emptyset$).
Now, I can't see why $Y$ and $X/\sim$ are not homeomorphic? Or at least isomorphic? I tried to draw those two sets in some cases and saw no difference.
Thanks for any help!
If you define: $$Y:=\{[x]\mid x\in X\}$$ and this set is equipped with the topology: $$\tau_Y:=\{U\in\wp(Y)\mid\cup U\in\tau_X\}$$ then $X/\sim$ and $Y$ are homeomorphic.
It is a nice way to "visualize" quotient spaces and gives you a good impression what it actually is.
The original set $X$ is partitioned and the elements of the partition become the elements of the quotient space.
If $U$ is a subset of the partition then it is open in $Y$ if and only if $\cup U:=\{x\in X\mid\exists u\in U[ x\in u]\}$ is open in $X$.