One thing I think is interesting about Spivak's book A Comprehensive Introduction to Differential Geometry is that Spivak uses metric spaces instead of topological spaces when defining an abstract manifold. For example, on p. 19:
A manifold-with-boundary is a metric space $M$ with the following property: If $x \in M$, then there is some neighborhood $U$ of $x$ and some integer $n \geq 0$ such that $U$ is homeomorphic to either $\mathbb R^n$ or $\mathbb H^n$.
Munkres takes a similar approach in the final chapter of Analysis on Manifolds.
In the preface to the first edition of A Comprehensive Introduction to Differential Geometry, Spivak states:
An acquaintance with topological spaces is even better, since it allows one to avoid the technical troubles which are sometimes relegated to the Problems, but I tried hard to make everything work without it.
What are the "technical troubles" that Spivak is referring to here? And why did Spivak have to try hard to make everything work out with this approach?
It seems like building the theory using the metric space definition would be no more difficult than building the theory using the topological space definition. I don't see what extra difficulties would arise.