Any reference that explains generated equivalence relation

Im looking for a reference that characterizes when an equivalence relation can be generated from a relation and gives a clear explanation of it.

Solution 1:

Recall that the equivalence relation generated by a relation is the smallest equivalence relation containing it. Equivalently, the equivalence relation $S$ generated by a relation $R$ is such that $(x,y)$ is in $S$ if and only if there exist $n\geqslant0$ and $(x_k)_{0\leqslant k\leqslant n}$ such that $x= x_0$, $y = x_n$, and for every $1\leqslant k\leqslant n$, either $(x_{k-1},x_{k})$ or $(x_{k},x_{k-1})$ belongs to $R$.

Since every equivalence relation is generated by itself, for every equivalence relation there exists a relation generating it.

To get $R\subseteq S$ generating $S$, one can erase from $S$ some or all of the following: (1) every $(x,x)$, (2) either $(x,y)$ or $(y,x)$ for every $x\ne y$ such that $(x,y)$ is in $S$, (3) every $(x,y)$ such that $(x,z)$ and $(z,y)$ are in $S$ for some $z\ne x,y$.

Solution 2:

No one actually gave a reference so I'll add one.

The book Introduction to topological manifolds by John M. Lee explains this at some length in the appendix A. Also, he gives good examples of generated equivalence relations in Chapter 3, Quotient Spaces.