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Let $\varphi: [a,b] \rightarrow \Bbb R^3$ be a continuous path and $a<c<d<b$.

Let $C =\{\varphi(t) \mid c \le t \le d\}$.

Must $\varphi^{-1}(C)$ be path-connected?

for above question, the answer is not-path-connected. However, I can't imagine how the inverse could not be path-connected. Any counter-example or proof that the inverse is not path-connected?

Fred Akalin
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delog
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1 Answers1

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Let $\varphi$ be a path that goes from a point to another point, and then returns, and choose $c$ and $d$ to be on the first portion of the path (not the return). Do you see why $\varphi^{-1}(C)$ is then not path-connected?

Fred Akalin
  • 730