No. of possible dense subsets of a metric space

Solution 1:

It must be that $X$ is almost discrete. Because if $p$ is not an isolated point of $X$, $C(p) = X \setminus \{p\}$ is open and dense. The intersection of finitely many open dense subsets is open and dense as well.

If $X$ has one non-isolated point $p$, then $X$ and $C(p)$ are the only dense subsets (as every dense subset must contains all isolated points, and those are $C(p)$). So this does not qualify.

So if $X$ has two non-isolated points $p \neq q$, then $X$, $C(p)$, $C(q)$ and $C(p) \cap C(q)$ are the only dense sets. So 4 of them.

If $X$ has 3 non-isolated points $p,q,r$, then every dense set contains $X\setminus \{p,q,r\}$ (all isolated points) and we can add any subset of $\{p,q,r\}$ to get different dense subsets, so we have 8 of them.

So 4 is the only one that can occur, among your list. E.g. for the metric space $X = \{0\} \cup \{\frac{1}{n}: n = 1,2,3,\ldots\} \cup \{2\} \cup \{2 + \frac{1}{n}: n =1,2,3.\ldots\}$ as a subspace of the reals.