Exercise 13, § 1

Identifying $\mathbf{R}^n$ with $\mathbf{R}^{n-1} \times \mathbf{R}$, let $f$ be a continuous mapping of an open subset $A$ of $\mathbf{R}^{n-1}$ into $\mathbf{R}$, and let $S$ be its graph. Show that the subspace $S$ of $\mathbf{R}^n$ is homeomorphic to $A$, and that the complement of $S$ in the "cylinder" $A \times \mathbf{R}$ is a dense open set in $A \times \mathbf{R}$, and has exactly two components if $A$ is connected.