Exercise 9, § 1

A broken line $L$ in $\mathbf{R}^n$ ($n > 1$) (Exercise 6) is said to be simple if it is homeomorphic to the interval $I = [0, 1]$ of $\mathbf{R}$. It amounts to the same thing to say that there exists a piecewise linear homeomorphism $u$

(*) It can be shown that an open set in $\mathbf{R}^m$ is not homeomorphic to any subset of $\mathbf{R}^n$, if $n < m$.

of I onto L (Exercise 6). Show that if A is a connected open set in $\mathbf{R}^2$ and if $L \subset A$ is a simple broken line, then $A - L$ is connected (argue by induction on the number of segments which make up L, and use Exercise 6 of Chapter I, § 11).