# Differential Geometry without General Topology

I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: "oh, but what's the problem ? First learn General Topology, and you'll understand Differential Geometry even better!" and I agree with that, but my point is: I'm a student of Physics, however I like to do everything with rigorous math, and of course I have special interested in math made with rigour like Spivak does, however, in the course of Physics we don't have General Topology and in the moment because of some things I'm studying I'm needing Differential Geometry, so I don't have time at the moment for General Topology.

Of course I'm interested in General Topology and of course in the future I'll study it, revisit Differential Geometry and make it more general, but for now I'm looking for some place which teaches without this. I'm looking for some treatment of differential geometry that defines manifolds without resorting to topological spaces and that gives good examples of constructing atlases.

Today my background is: basic set theory, single and multivariable calculus, ordinary differential equations, linear and multilinear algebra, a little bit of abstract algebra and also the basic topology of $\mathbb{R}^n$. Also I know the construction of vectors as derivations in euclidean spaces, as well as the definitions of tangent and cotangent spaces in the euclidean space.

Is there some book that teaches differential geometry (starting from the definition of manifold) in a way that it's suitable for someone who has this background?

Thanks for your help, and sorry if this question is silly, or if it's not suitable here.

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I deleted my answer because I realised this isn't exactly what you want, but you still might have some interest in it, so I leave it as a comment. – Git Gud Apr 4 '13 at 21:28
What sort of differential geometry you want? Curves and surfaces one can do with a bare minimum of knowledge of the topology of $\mathbb R^n$. If you want manifolds and the whole package, well, learn a big of general topology: it is unavoidable. – Mariano Suárez-Alvarez Apr 4 '13 at 21:36
Two downvotes? Just what is wrong with this question? – Git Gud Apr 4 '13 at 21:38
Maybe Do Carmo, Differential geometry of curves and surfaces ? – Vincent Nivoliers Apr 4 '13 at 22:01
Gauss didn't know anything about topology, and he did OK. – bubba Apr 5 '13 at 0:00

By basic topology of $\mathbb{R}^n$'' I assume that you are familiar with the notions of openness, closedness, connectedness, and compactness. If you are unclear on these notions (I found compactness hard to get used to), you should remedy that before attempting to learn differential geometry.