The Quotient Topology of a Discrete Topological Space

analysisconnectednessequivalence-relationsgeneral-topologyquotient-spaces

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 $\widetilde{\mathbb{X}}$.

  1. Does $\tau_{\widetilde{\mathbb{X}}}$ have to be the discrete topology on $\widetilde{\mathbb{X}}$?
  2. Is $\tau_{\widetilde{\mathbb{X}}}$ a hausdorff topology?
  3. 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 $x\not\sim y$.

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