**Problem: How many dense subsets disjoint, totally disconnected in pairs $ mathbb {R} $ have (in the standard topology)?**

This is how I did it. I am not sure if this is correct or not. I know I have the idea, but I can't say positively if this will work or not.

Leave $ p in mathbb {R} $such that $ p notin mathbb {Q} $that is to say $ p in mathbb {Q ^ {C}} $. So, let's consider 2 open sets $ S $ Y $ T $such that $ S = mathbb {Q} cup $ {p} and $ T = mathbb {Q ^ {C}} $ {P}. So both $ S $ Y $ T $ they are dense in $ mathbb {R} $. Too, $ S cap T $ = $ emptyset $that is to say $ S $ Y $ T $ they are disjoint so I guess we can show that $ S $ Y $ T $ they are both a totally disconnected subset of $ mathbb {R} $ as previously. can i say that $ (S, T) $ is a pair, such that $ S $ Y $ T $ are disjoint and both are dense in $ mathbb {R} $ and disconnected But I'm not sure what arguments and reasoning to show to establish this claim.

As, $ p in mathbb {Q ^ {C}} $ is arbitrary and $ mathbb {Q ^ {C}} $ is infinite, therefore we can infinitely construct many of those pairs.

Can anyone help me on this and show explicitly? I appreciate your help and support in this.