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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?

Asaf Karagila
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The homeschooler
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2 Answers2

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You need to be acquainted with the https://proofwiki.org/wiki/Fundamental_Theorem_on_Equivalence_Relations and its converse.

Every equivalence relation partitions the set it is on into disjoint subsets. Each of the elements in a given subset is equivalent to all the other elements in that subset. They divide the population into "bubbles", if you like.

The set of all these subsets is the quotient set induced by that equivalence relation.

Prime Mover
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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).

GReyes
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    A partition of $S$ is by definition a family $\mathscr{P}$ of pairwise disjoint, non-empty subsets of $S$ whose union is $S$. The set of equivalence classes of an equivalence relation on $S$ is exactly that: the equivalence classes are subsets of $S$, they are non-empty and pairwise disjoint, and their union is all of $S$. (You may find this answer of some use.) – Brian M. Scott Feb 06 '21 at 00:11