@Adeek You can find disjoint balls with centers in the points of $A$. You have uncountably many of them. If $D$ is dense in $X$, then each of this balls contains at least one point from $D$. So $D$ is uncountable.
I guess this is quite a common question so that are probably several posts about his on the main site.
assuming that result you posted above. if I consider for example $l_p(\gamma)$ how can I find f,g that satisfy this condition ? I mean the domain that we are talking about is arbitrarily uncountable set ?
BTW there is also general topology room. Your question would fit nicely in the topic of the room. However, for quite a long time there is almost no activity in that room, so it might go unnoticed if you post there. But let's hope that the general topology room will become active later again.
And also I like the result that each Hilbert space is isomorphic to $\ell_2(\Gamma)$ for some $\Gamma$. (Although I do not know about some reasonable application of this results. But still, knowing that they are completely characterized and that that there is one simple cardinal invariant which determines whether two such spaces are isomorphic is kind of nice.)
