Hausdorff Quotients:

The quotient of a Hausdorff space need not be Hausdorff. Since quotient spaces occur in abundance we need easily verifiable sufficient conditions for a quotient space to be Hausdorff. We provide here one such condition which suffices for most applications [16]. Let $X$ be a space on which an equivalence relation $\sim$ has been defined. Note that a relation on $X$ is a subset $\Gamma$ of the cartesian product $X\times X$ on which we have the product topology. Thus,

$$\Gamma = \{(x, y)\in X\times X\;/\; x\sim y\}$$