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The quotient set $X/\sim$ of a set $X$ by an equivalence relation $\sim$ is the set of its equivalence classes.

A quotient topological space identifies points of a topological space according to an equivalence relation and equips the set of equivalence classes with the quotient topology. A subset is open exactly when its inverse image under the canonical quotient map is open. Even a Hausdorff space can have a non-Hausdorff quotient; identifying real numbers differing by a rational number gives a multi-point quotient with the indiscrete topology.

A union of equivalence relations on the same set is reflexive and symmetric, but can fail transitivity. On a three-point set, partitions into blocks $\{1,2\},\{3\}$ and $\{1\},\{2,3\}$ make the union relate $1$ to $2$ and $2$ to $3$ without relating $1$ to $3$. An equivalence relation containing both requires transitive closure as well.