Topologist's sine curve

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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. topologist's sine curve

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: