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What kind of entry level?
For point-set topo I guess Topology by James Munkres was pretty good; I don't know, the class that used that book was kind of a mess
Algebraic Topology I would say Algebraic Topology by Allen Hatcher.

I haven't really taken real analysis other than Rudin, nor have I actually done any Lie groups/algebras formally; for that stuff I basically just use bits of diff. geom. and Wikipedia.
Professor Georgi here wrote a book on Lie algebras in Particle Physics, but while Georgi is a very brilliant physicist and has a wonderful pedagogical model, he's hard to understand both by physicists and mathematicians because his intuition is very strange.

Well, right now for real analysis I'm using Stein and Shakarchi (the class text, since Stein is a professor here), but reading out of Papa Rudin as well. I'm looking for a book that's about as explicit as Rudin, when it comes to proofs, but is a bit easier to read. The class I'm taking is covering abstract measures and Lebesgue integration, Hilbert spaces, and Hausdorff dimension. I want to be super-duper confident in methods of proof in basic measure theory and integration before we hit the other stuff.

For topology, I have Shick's Topology: Point-Set and Geometric, which while easy to understand is super-duper basic and isn't really giving me anything more than I already knew. I'll check out Munkres, I think that's what the class here uses. I guess here I'm just looking for something that'll get me set up for algebraic topology.

For that, I'm using the Graduate Texts book by Rotman. It's a bit hard to read through, but it's a pretty good book. I'll check out Hatcher. I'm doing my independent seminar in low-dimensional topology, so in both a point-set and algebraic topology book, I'm looking for something that gets me used to the terminology and methods of proof used in this field. In particular, I want something that gets me through Thurston's "Three-dimensional Geometry and Topology" and Carter et al's "Surfaces in 4-Space".

As far as Lie groups/algebras go, all I have are things I've gotten from Do Carmo's Riemannian Geometry. It tends to lack actual examples of anything. After I'm done with it, I'll probably be looking for a better book on differential geometry.

Thanks for the responses, guys!

I feel the same way about Real Analysis. Sadly, I had to drop it. (Well, not too sadly...I hate analysis). I guess I'll just have to settle for the lower-level version next semester.

That is very nice. I have a great deal of respect for that company's work, and the company itself seems to be a good place to be.

Do you know what you will be doing for them?

I've been trying to piece the puzzle together but I've lost track... Whereabouts on the educational ladder are you, out of curiosity?

And Layra I envy you. Or at least I would if I didn't have an aversion to that area of Mass due to traumatic childhood clothes shopping memories.

I'm not earning my money... I'm being 'funded' by the HEA and extorting desperate students during exam time.

I'm a junior in college, and a math major. I just happen to have been *very* ill-prepared for math (at least in my eyes), considering I'm just now getting to real analysis.

heh... No real story to tell unfortunately. Just memories of being in a mental state of sheer boredom and desperation as I was dragged through various shops and plazas. But now I'm older and thousands of miles away so I'm safe ninja

I did essentially the same thing you did, I guess, plus a bunch of random Lebesgue measure and integration theory as well as some bits of Hilbert spaces; I don't know, it was rather unconnected so I didn't really know what was happening all the time.