Passing to the quotient (topology)

See passing to the quotient for other forms of this theorem

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Statement

$\xymatrix{ X \ar[d]_\pi \ar[dr]^f & \\ \frac{X}{\sim} \ar@{.>}[r]^{\overline{f} }& Y}$
$f$ descends to the quotient

Suppose that $(X,\mathcal{ J })$ is a topological space and $\sim$ is an equivalence relation, let $(\frac{X}{\sim},\mathcal{ Q })$ be the resulting quotient topology and $\pi:X\rightarrow\frac{X}{\sim}$ the resulting quotient map, then:

• Let $(Y,\mathcal{ K })$ be any topological space and let $f:X\rightarrow Y$ be a continuous map that is constant on the fibres of $\pi$ then:
• there exists a unique continuous map, $\bar{f}:\frac{X}{\sim}\rightarrow Y$ such that $f=\overline{f}\circ\pi$

We may then say $f$ descends to the quotient or passes to the quotient

Note: this is an instance of passing-to-the-quotient for functions

This is an instance of passing to the quotient (function) with functions between sets. In this proof we apply this theorem and show the result yields a continuous mapping (by assuming both $f$ and $\pi$ are continuous)