So I've just started trying to teach myself some topology and i cant quite grasp how an equivalence class forms a partition more specifically i don't understand the proof that there is exactly one equivalence relation that forms the partition.
For example in my book it says that a partition $D$ is formed from an equivalence relation $R$ and that equivalence relation is unique that is there is only one equivalence relation that forms the partition however they do not give a proof of this i myself cant come up with one so is there a proof of this statement?
Thanks in advance.
Best Answer
Suppose $R_1$ and $R_2$ on a set S induce the same partition of S but are not identical.
Since they are not identical there is ( at lest) one couple $(a,b)\in S\times S$ that belongs to one of them but does not belong to the other. [ Relations are sets - of couples - and two different sets must differ by at least one element, by the extensonnality principle]
So, one relation will produce a partition having as "cell" an equivalence class with $a$ and $b$ as elements; while the other relation will not produce such a partition.
The reason is that in the other relation there is no couple $( a,b)$ , meaning that there is no equivalence class having $a$ and $b$ as members. Since the partition is a set of equivalence classes, and since the two partitions differ by ( at least) one equivalence class, they are not identical.
The fact that the partitions induced are not identical contradicts our assumption.
So $R_1 = R_2$.