Quotient topology/Mapping to a set definition

Grade: A

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Definition

Let $(X,\mathcal{J})$ be a topological space and let $h:X\rightarrow Y$ be a surjective map onto a set $Y$, then the quotient topology, $\mathcal{K}\subseteq\mathcal{P}(Y)$ is a topology we define on $Y$ as follows:

• $\forall U\in\mathcal{P}(Y)[Y\in\mathcal{K}\iff h^{-1}(U)\in\mathcal{J}]$ or equivalently:
• $\mathcal{K}=\{U\in\mathcal{P}(Y)\ \vert\ h^{-1}(U)\in\mathcal{J}\}$

The quotient topology on $Y$ consists of all those subsets of $Y$ whose pre-image (under $h$) is open in $X$