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I am a newbie to topology. Hope someone can help sharpen my understanding.

I read a theorem that "if [tex] T [/tex] is a topological space with countable base, then every open cover has a finite or countable subcover"

Apparently this is quite intuitive because as in the definition of countable base, every open set should be generated by it, and [tex] T [/tex] is by definition open, so that means it should be the union of the countable base. Obviously this is the "smallest size" open cover which is of course countable, so any open cover cannot get smaller "size" than this?

Also, somehow I am quite confused with countable subcover, because if an open cover is "as countable as" the base, this means every set is just that in the base, hence, this is the just the open cover formed by the base, so is the above theorem valid, i.e. open cover and subcover are the same is allowed?

Thanks.

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# Countable base and subcover

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