3.02 Quotient topology: continuous maps

We show that the continuous maps out of a quotient space are precisely the continuous maps before taking quotient which descend to the quotient. For notes, see here: www.homepages.ucl.ac.uk/~ucahj...

[@laflaca5391]

Great series, keep on!

[@juaniribasdelapuerta9535]

Just to make sure, the proof of the existance of Fbar given the fact that F depends only on the equivalence classes follows from the following construction: for every A in X/~, we take "a" in A, and define Fbar(A) as F(a). The construction is consistant as if we had took any other "b" in A, as we would then have a~b, therefore F(b)=F(a)=Fbar(A) by hypothesis. That's the idea?

[@jonathanevans27]

Yes, that's why Fbar exists as a map of sets. The point is then that it is continuous. This works because the quotient topology is defined to make it continuous.