Differential Geometry without General Topology

Solution 1:

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.

If you understand these, then you're probably already prepared to read an introductory book on differential geometry, such as do Carmo's Differential Geometry of Curves and Surfaces or O'Neill's Elementary Differential Geometry. Apart from the concepts I mentioned above, all the necessary topology is developed alongside the geometry in these books (e.g. homeomorphism, homotopy, Euler characteristic, and so on).

If you want to learn quickly about the topology of smooth manifolds without having to learn about general topological spaces, there is probably no better place to look than Milnor's Topology from the Differentiable Viewpoint. A more in-depth treatment along the same lines is Guillemin and Pollack's Differential Topology.