I am sorry, if this is a repetition of previous questions. But my case is sightly different. I am a physics undergrad who wants to shift to pure maths, and I want to study topology. The supreme reference is apparently Munkres , but I think it would be too much of a time-investment to study point-set topology from it. I want a shorter treatment of point-set topology, so that I can quickly move on to algebraic topology. Two of the books which I am thinking about are Armstrong's 'Basic Topology' and Lee's 'Topological manifolds'. Do you think it could give a shorter more effective treatment. I am not completely new to topology, and have been exposed to it before in Physics. I understand the intuitive meaning of quotient spaces, compactness, topological groups, definition of fundamental group and homology.
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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:
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.
I think the perfect book for you would be Glen E. Bredon's "Topology and Geometry". It has a clear introduction to smooth manifolds as well. I use this book a lot and highly recommend it.