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I need to prove that if X has a countable base, then every base contains a countable base. I understand that this is a duplicate of another stack exchange question, but I do not know why a given element of the countable base can be expressed as the union of elements of the possibly uncountable base.
I thought that the definition of base simply says that every element of the topology can be written in this way, but not necessarily every subset of the topology.

HBHSU
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    Of course, if B is a countable, then B cantains the countable base B. The problem is vacuous. Did you read the problem correctly? If so, do not go guessing, ask the source. – William Elliot Oct 29 '17 at 04:55
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    The problem as you have stated it makes no sense. Are you sure you copied it correctly? I will take a stab in the dark and guess that you were asked to prove something like this: if the space has a countable base (call it $\mathcal A$) and if $\mathcal B$ is some other (not necessarily countable) base for the same space, then there is a countable base $\mathcal C$ which is contained in $\mathcal B.$ Anyway, it's a true statement, and it's a good exercise for a beginning class in general topology, and it uses some of the same words that you used in your question. – bof Oct 29 '17 at 05:00

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