Given a surjective function $p$ that takes a space $X$ to it’s decomposition $X*$ (decomposition here means a partition into equivalence classes), is it always the case that the function $p$ is a quotient map? And why is there only one unique topology (the quotient topology) on $X*$ that ensures $p$ is a quotient map?
Relation between quotient map and quotient space
general-topologyquotient-spaces
Best Answer
p is a quotient map only if you define the topology of X* to be the final topology generated by p, the largest topology that makes p continuous which is
{ V : p$^{-1}$(V) open set of X }