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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"?

Arctic Char
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Chris Ewing
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    This thread has some relevant comments: https://math.stackexchange.com/q/3547427/40119 User qwertyguy commented "This property [having a countable basis] is not universially required though: it ensures the existence of the so called 'partitions of unity', which are a fundamental tool for the development of the theory (but many authors drop this assumption)" – littleO May 17 '21 at 02:10
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    It’s a matter of taste. Both are accepted and neither is wrong. – Randall May 17 '21 at 02:59

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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).

Son Gohan
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