Now I have

, which is, of course, continuous (given the prior statements) from

.

So if

denotes the canonical quotient map from

to

, I would like to say that

where

is the desired induced map. However, it remains to be shown (in order to apply the theorem I'm basically quoting here, Theorem 22.2 in Munkres) that

is constant on each set

, and this is where I am stuck. Looks like this means that

takes a set of things that are equivalent in

(that's

, of course) and says that these are also equivalent under the quotient map on

. I can't tie it together.

I've either got the wrong approach, or I could use some intuitive enlightenment. This whole set-up seems to imply some necessary connection between the quotient maps/spaces, but I'm not seeing the "necessitude".