I've been studying Differential Geometry on Spivak's Differential Geometry book. Since Spivak just works with notions of metric spaces and analysis, I'm doing fine. The point is that Spivak presents the following definition of a manifold:
A manifold $M$ is a metric space such that for every $p \in M$ there's some neighbourhood $V$ of $p$ and some integer $n \geq 0$ such that $V$ is homeomorphic to $\Bbb R^n$
Now, there's another definition, usually given in texts that assume the reader knows general topology, and the definition is:
A manifold $M$ is a topological space such that:
- $M$ is Hausdorff;
- $M$ has a countable basis for its topology;
- $M$ is locally Euclidean.
For now I'm happy with Spivak's definition because I've not seen general topology yet, but I'm curious with one thing: these two definitions are equivalent? In other words, every topological space with those three properties is metrizable, so that it can be put in terms of the first definition? Is there any other way in which these definitions can be said to be equivalent?
Thanks very much in advance for the help.