Question of topology

Question

Is it true that the image of nowhere dense set under a continuous mapping is nowhere dense?

Answer 1

This is false. Consider $f: \mathbb{R}\to \{0\}$ by $f(x) = 0$, $\forall x\in\mathbb{R}$. Note that $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$ but you are mapping to the entire topological space.

Answer 2

Very false. The (uniformly) continuous Cantor function from $[0,1]$ to $[0,1]$ maps the nowhere dense Cantor set onto $[0,1]$. In fact we can have any compact metric space $X$ as the image of such a continuous map from a nowhere dense set. The Cantor fucntion is just a convenient concrete example of this.