Is $U$ a connected subset of $X$?

Question

Let $X$ be a connected topological space. Let $U$ be a dense subset of $X.$ Can we say that $U$ is also connected?

Any help in this regard will be highly appreciated. Thank you very much.

Answer 1

Title question: yes, if $U$ is a dense subset of $X$, then $U$ is a dense subset of $X$.

Other question: no, see for instance $\Bbb R\setminus\{0\}$ in $\Bbb R$.