Difference between revisions of "Characteristic property of the quotient topology/Statement"

Revision as of 17:22, 9 October 2016

In this commutative diagram $f$ is continuous $\iff$ $f\circ q$ is continuous
$\xymatrix{ X \ar[d]_{q} \ar[dr]^{f\circ q} & \\ Y \ar[r]_f & Z }$

Let

$(X,\mathcal{ J })$ and

$(Y,\mathcal{ K })$ be topological spaces and let

$q:X\rightarrow Y$ be a quotient map. Then^[1]:

For any topological space, $(Z,\mathcal{ H })$ a map, $f:Y\rightarrow Z$ is continuous if and only if the composite map, $f\circ q$ , is continuous