Question on Spivak's Definition of a Manifold. I am reading the first volume of Spivak's Differential Geometry Series, A Comprehensive Introduction to Differential Geometry, and I don't understand his definition of a manifold.
He says that

A metric space $M$ is a manifold if for each $x\in M$, there is some neighborhood $U$ of $X$ and some integer $n\ge 0$ such that $U$ is homeomorphic to $\mathbb{R}^n$.

I understand how the (usual) Hausdorff condition of a manifold comes from the fact that $M$ is a metric space in this definition (of course, one can get rid of the metric space condition and just let $M$ be Hausdorff or just a topological space), but how does Spivak's definition account for the condition that $M$ must be second countable?
Thank you for your help.
 A: Short answer: the definition without "second countable" is not equivalent to the definition with "second countable".
For a longer answer, read the rest of that page, and follow the instructions to read Appendix A. After reading Appendix A, you will hopefully have a better understanding of how the definition of manifold is not written in stone, and that when one writes about manifolds one should be precise regarding which of the various topological properties one is assuming (e.g. metrizable; Hausdorff; second countable; paracompact; ...)
A: It doesn't. The given definition could include non-second countable spaces, like $\mathbb{R}^2$ with the metric $$d((x_1,x_2),(y_1,y_2)) = \begin{cases}|x_2-y_2|& \text{ if } x_1=y_1,\\ 1 & \text{ otherwise}\end{cases},$$ which is homeomorphic to the disjoint union of uncountably many copies of $\mathbb{R}$.
A: In his appendix A, Spivak defines a manifold as a locally euclidean Hausdorff space (i.e. same as in chapter 1 but with "metric" replaced by "Hausdorff") and proves the following theorem:

The following are equivalent for any manifold M:
(a) Each component of M is σ-compact.
(b) Each component of M is second countable.
(c) M is metrizable.
(d) M is paracompact.

