My case was similar to yours: I graduated in theoretical physics but then made the transition to pure mathematics, so I had to quickly get a good grasp of required background material. The fastest path I know to the essential point-set/metric topology needed to start algebraic topology is the recent textbook:
- Runde, V. - A Taste of Topology.
It develops all elementary concepts and proves all standard theorems in just ~165p. in a course-like set of rigorous lectures with exercises. I think it is the best supplement of, or starting point before, Bredon's "Topology and Geometry", as this last title is geared towards algebraic topology and develops general and differential topology in a very succinct manner (although very complete!). That couple of books would make a quick route to what you want. You can check out other suitable book collections at my Amazon listmanias.
Chapter 1 of Hatcher corresponds to chapter 9 of Munkres. These topology video lectures (syllabus here) do chapters 2, 3 & 4 (topological space in terms of open sets, relating this to neighbourhoods, closed sets, limit points, interior, exterior, closure, boundary, denseness, base, subbase, constructions [subspace, product space, quotient space], continuity, connectedness, compactness, metric spaces, countability & separation) of Munkres before going on to do 9 straight away so you could take this as a guide to what you need to know from Munkres before doing Hatcher, however if you actually look at the subject you'll see chapter 4 of Munkres (questions of countability, separability, regularity & normality of spaces etc...) don't really appear in Hatcher apart from things on Hausdorff spaces which appear only as part of some exercises or in a few concepts tied up with manifolds (in other words, these concepts may be being implicitly assumed). Thus basing our judgement off of this we see that the first chapter of Naber is sufficient on these grounds... However you'd need the first 4 chapters of Lee's book to get this material in, & then skip to chapter 7 (with 5 & 6 of Lee relating to chapter 2 of Hatcher).
There's a crazy amount of abstract algebra involved in this subject (an introduction to which you'll find after lecture 25 in here) so I'd be equally worried about that if I didn't know much algebra.
These video lectures (syllabus here) follow Hatcher & I found the very little I've seen useful mainly for the motivation the guy gives. If you download the files & use a program like IrfanView to view the pictures as you watch the video on vlc player or whatever it's much more bearable since you can freeze the position of the screen on the board as you scroll through 200 + pictures.
I wouldn't recommend you treat point set topology as something one could just rush through, I did & suffered very badly for it...
Best Answer
This is a very non-formal (and not many mathematicians would use that wording) way of simply saying that $X=S^2\vee S^2$ where $\vee$ is the wedge sum operator a.k.a. glueing at a point. Or in other words these are two spheres touching each other at exactly one point:
As you can see from the image these are two "copies" of $S^2$.