TITLE: What are the "technical troubles" with using a metric space rather than a topological space when defining an abstract manifold? (As in Spivak)
QUESTION [10 upvotes]: 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.

REPLY [4 votes]: The main reason for not including a metric in the definition of differential manifold is that it is an irrelevant artificial structure which is immediately abandoned when one defines an atlas: obviously transition maps between charts are smooth but definitely not isometric!
In  the same vein the category of  smooth manifolds and that of metric spaces are not related by natural functors in either direction.
A more technical issue is that some manifolds are plain not metrizable, even in dimension one: Spivak himself describes an example, the long line (Appendix A, corollary 6).
It is true that such manifolds are not very common, but it is very unaesthetic to exclude them from the very definition, especially since you might start with metrizable manifolds and after operating on them with standard constructions end up with a non-metrizable one.