Metric Spaces needed for Differential Geometry

I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and Surfaces and O'neill, however I've heard that Spivak's "A Comprehensive Introduction to Differential Geometry" indeed doesn't assumes general topology.

Spivak just asks that the readers knows about his calculus on manifolds and basics of metric spaces. I've already studied his "Calculus on Manifolds" and I've also studied basic topology of euclidean spaces, however about metric spaces I know just the really basic things: the definition, the facts of topology of euclidean spaces stated for metric spaces just replacing the metric that defines the balls, subspaces, and what are homeomorphisms.

Indeed with just that I managed to understand the first chapter of Spivak's book pretty easily. The point is: is all of that the necessary knowledge of metric spaces to understand Spivak's book? Is there some reference where I can look for that "basic knowledge" needed? Maybe some notes.

Thanks a lot in advance, and sorry if this question is silly.