Let $Z=\{(x,\sin(\frac{\pi}{x})\mid 0<x\leq1\}$, and suppose $X=Z\bigcup\{(0,0)\}$ is equipped with the subspace topology. Show that any path in $X$ starting from $(0,0)$ must be the constant path.
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A path is a continuous function from $[0,1]$ to the set $Z$. More importantly, the image of the path is a compact connected subset of $Z$. Now, think of what compact connected subsets of $Z$ contain $(0,0)$. – Sarvesh Ravichandran Iyer May 21 '19 at 04:49
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1Minor variant of Topologist's sine curve is not path-connected – YuiTo Cheng May 21 '19 at 05:20