Exercise 13, § 11

Let $\theta$ be an irrational number. Let $f$ be the mapping of the real line $\mathbf{R}$ into the torus $\mathbf{T}^2$ for which the image $f(t)$ of each $t \in \mathbf{R}$ is the canonical image in $\mathbf{T}^2$ of $(t, \theta t) \in \mathbf{R}^2$. $f$ is injective and continuous. Show that, in the topology on $\mathbf{R}$ which is the inverse image under $f$ of the topology of $\mathbf{T}^2$, $\mathbf{R}$ is connected but no point of $\mathbf{R}$ has a fundamental system of connected neighbourhoods.*