I'm working on a problem in Armstrong's Basic Topology and have hit a wall. Exercise 35 on page 41 reads:
Given a map $f : X \to \mathbb{E}^{n+1} \setminus \{0\}$ find a map $g: X \to S^n$ which agrees with $f$ on the set $f^{-1}(S^n)$.
My hunch is that this is an extension problem. In any case, I guess I've missed the discussion in the text about these codomains, and I'm unclear about how the spaces $\mathbb{E}^{n+1}$ and $S^{n}$ relate to one another. In fact, I'm really not certain what's being asked.
Can anyone direct me to a good source to read up on these spaces and the kinds of maps that behave nicely between them?
This is a homework question, so please no direct answers. Hints that point me to the heart of the material, however, are certainly appreciated.