Let $(\mathbb{X},\tau_{\mathbb{X}})$ be a discrete topological space.
Suppose that an equivalence relation $\sim$ on $\mathbb{X}$ is defined.
Let $\widetilde{\mathbb{X}} := \mathbb{X} / \sim$ be the quotient space and $\pi: \mathbb{X} \longrightarrow \widetilde{\mathbb{X}}$ the projection map.
Let $\tau_{\widetilde{\mathbb{X}}}$ be the quotient topology on $\widetilde{\mathbb{X}}$.
- Does $\tau_{\widetilde{\mathbb{X}}}$ have to be the discrete topology on $\widetilde{\mathbb{X}}$?
- Is $\tau_{\widetilde{\mathbb{X}}}$ a hausdorff topology?
- Is $(\widetilde{\mathbb{X}},\tau_{\widetilde{\mathbb{X}}})$ connected?
Best Answer
The quotient topology on a given space $X$ is defined in the following way: a subset $U\subseteq X/\sim$ is open if and only if $\pi^{-1}(U)$ is open in $X$. In your case $X$ is discrete and so every subset of $X$ is open. In particular $\pi^{-1}(U)$ is open for any subset $U\subseteq X/\sim$, meaning $U$ is open in $X/\sim$ and therefore $X/\sim$ is discrete as well.
This implies that $X/\sim$ is Hausdorff. And disconnected as long as it has at least $2$ points, meaning as long as there are two points $x,y\in X$ such that $x\not\sim y$.