Quotient topology/Equivalence relation definition

Grade: A

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Definition

Given a topological space, $(X,\mathcal{J})$ and an equivalence relation on $X$, $\sim$, the quotient topology on $\frac{X}{\sim}$, $\mathcal{K}$ is defined as:

• The set $\mathcal{K}\subseteq\mathcal{P}(\frac{X}{\sim})$ such that:
  • $\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}]$ or equivalently
• $\mathcal{K}=\{U\in\mathcal{P}(\frac{X}{\sim})\ \vert\ \pi^{-1}(U)\in\mathcal{J}\}$

In words:

• The topology on $\frac{X}{\sim}$ consists of all those sets whose pre-image (under $\pi$) are open in $X$