Topologist's sine curve
The topologist's sine curve is the closure in $\mathbb{R}^2$ of the graph of the function
$$f(x) = \left\{ \begin{array}{ll}
\sin \left( \dfrac{1}{x} \right) &; x>0 \\
0 &; x=0
\end{array} \right.$$
The space itself consists of the graph of the above function along with the set $\{(x,y) \colon x=0, -1 \leq y \leq 1\}$ called the limit bar.
Properties
Theorem: The topologist's sine curve is irreducible between points $(0,y)$ and $(1,\sin 1)$ for all $-1 \leq y \leq 1$.
Proof: █
Theorem: The topologist's sine curve is a compactification of the ray $(0,1]$ with remainder an arc.
Proof: █
Theorem: The hyperspace of all subcontinua of the topologist's sine curve is homeomorphic to the cone over the topologist's sine curve.
Proof: █
