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radou

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So, let X be a topological space which satisfies the second axiom of countability, i.e. there exist some basis B such that its cardinal number is less or equal to [tex]\aleph_{0}[/tex]. One needs to show that such a space is Lindelöf and separable.

To show that it's separable, let B be a countable basis for X. From every element of the basis, take one element x, and we obtain a countable set S = {x1, x2, ...}. Now we only need to show that this set S is dense in X, and this is true if and only if its intersection with every open set in X is non-empty. So, let U be some open set in X. Clearly, U can be written as a union consisting of the basis sets, and its intersection with S is definitely non-empty.

To show that X is Lindelöf, let C be some open cover for X. The countable basis B is, by definition, an open cover, too. We only need to show that B is a refinement of C, which is obvious, because for every set U in C has a subset which belongs to B (since it can be written as a union of the basis sets).

I hope this works, thanks in advance.