Confusion about the definition of a manifold In many books I have seen the definition of a topological manifold as follows
A topological manifold is a topological space that is Hausdorff, has a countable basis and is locally Euclidean.
But in other definitions the countable basis part is omitted, so which of the two is true or "correct"?
 A: In primis, I would follow the suggestion of Randall that as long as we talk about topological manifolds, it’s a matter of tastes, that is if we allow certain strange topological spaces with uncountable basis to be called topological manifolds or not.
But as long as we develop the theory of manifold to the differentiable and (then) the metric point of view, one definitely wants to assume the countable basis assumption. Indeed assuming it one can show that all the assumptions become equivalent to paracompactness of the manifold, and this assures you can have a wonderful friend in your toolkit: partition of unity.
Indeed, partition of unity is essential to develop all the theory of integration on manifolds, reaching the famous and elegant Stokes’ theorem. Moreover, partition of unity is the fundamental tool also for proving that is without loss of generality to assume that every manifold comes equipped with a smooth scalar product on each tangent space (which is the starting point of Riemannian Geometry).
