See Theorem 19.6 of Munkres if you need a detailed proof. Note that an identity map is continuous.

Figure 1.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".

Figure 2.

To apply theorem 22.2 of Munkres, we need to show that for is constant by a mapping g in Figure 1. If t is not {0} or {1}, each is a single point in , so it is straightforward to see that g is constant on those points (See Figure 2)

By the definition of the suspension map, we see that or is mapped by p to a single point in . We call it "u" and "v", respectively. To apply theorem 22.2 of Munkres, we need to show that and are constant within the mapping of g. Note that is mapped by to ( to , respectively) (See Figure 2). Then they are constant by a mapping q in Figure 2 because (or ) is mapped to a single point by a mapping q in Figure 2. Now the hypothesis of Theorem 22.2 of Munkres is satisfied and we conclude that is continuous.