So I've just started trying to teach myself some topology and im really confused on what a partition is and how it is in any way related to an equivalence relation.
My main confusion however is that of the concept of a quotient set I've read that a quotient set is the set of all equivalence classes of a set namely if $S$ is some set with an equivalence relation $R$
Then is the quotient set $S/R=\{[a] | a \in S\} $
My question is how is this quotient set the same as a partition of a set?
Best Answer
When you have a quotient, each class is a subset of elements related by whatever your relation $R$ is. Those subsets are disjoint and each element of $S$ belongs to one of them. Therefore they form a partition of $S$. To see that they are disjoint, suppose $x\in [a]\cap[b]$ with $a$ and $b$ not related by $R$. Then you would have $xRa$ and $xRb$ and from transitivity $aRb$, a contradiction. It should be clear that each element of $S$ belongs to some class (which could be a singleton).