Denseness in intersection

Question

Let $Y$ and $Z$ be topologycal spaces (if necessary, Banach spaces), and let $X$ be dense set in $Y$ and in $Z$.

Then, is $X$ dense set in $Y \cap Z$? If not, what assumption I need?

Answer 1

From the context I gather that $Y,Z$ are both subsets of some larger space which I'll call $S$ (or the intersection makes no sense) and $X$ is a subset of both $Y$ and $Z$.

$X$ being dense in $Y$ means that $X \subseteq Y$ and $Y \subseteq \overline{X}$ (closure in $S$) and similarly we know $Z \subseteq \overline{X}$. It follows that $Y \cap Z \subseteq \overline{X}$ and thus $X$ is dense in $Y \cap Z$.