Inverse function that takes connected set to non-connected set I've been struggling with providing examples of the following:
1) A continuous function $f$ and a connected set $E$ such that $f^{-1}(E)$ is not connected
2) A continuous function $g$ and a compact set $K$ such that $f^{-1}(K)$ is not compact

Take any space $X$ which is not connected and not compact. For example, you could think of $\mathbf R - \{0\}$. Map this to a topological space consisting of one point. [What properties does such a space have?]
