Density of a subset of a dense set

Question

Let $A$ be a dense set in $[0, 1]$ and let $B\subseteq A$ such that $B$ is not dense in $A$. What can be said about the density of $B$ in $[0, 1]$? Is there any result that may prove that $B$ is/isn't dense in $[0, 1]$?

Answer 1

If $B$ is not dense in $A$, there is a nonempty open subset $U$ in $A$ (with respect to the subspace topology) which contains no points of $B$. By definition of the subspace topology, $U = V \cap A$ for some open set $V \subset [0,1]$. Now $V$ is nonempty and contains no points of $B$, so $B$ cannot be dense in $[0,1]$.

Answer 2

By a standard formula for subspace closures:

$$\operatorname{cl}_A(B) = \operatorname{cl}_X(B) \cap A$$

for all $B \subseteq A$. So if $B$ is not dense in $A$, $\operatorname{cl}_A(B) \neq A$, so then $A \nsubseteq \operatorname{cl}_X(B)$ and thus $B$ is not dense in $X$ either.