I'm not very good in English, but I don't think is a cover...

I think you mean X is separable, and Q is countable and dense in X... I don't know if there is a english name for this kind of set...

First of all, you should note is countable:

Secondly is not a subcover of .

Thirdly, the idea is good, and covers X:

Let . Then for some . is neighbourhood hence there is the ball . Q is dense in X so there is such that . There exists ball where . Thus , and .

Of course . Hence (and too) is cover of X.