Let $f: [0,1] \to X$ be a continuous function from the segment $[0,1]$ to a Hausdorff space $X$ such that $f(0) \ne f(1)$.
Can we claim that there is always an injective continuous function $g: [0,1] \to X$ such that $g(0)=f(0)$ and $g(1)=f(1)$ ?
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Does $g$ need to have any relation to $f$ other than at endpoints $0,1$? – coffeemath May 03 '19 at 04:17
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@coffeemath No, only the properties listed above are mandatory. – John McClane May 03 '19 at 04:32
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3Possible duplicate of A question about path-connected and arcwise-connected spaces (The definition of arcwise connectedness is every two distinct points can be connected by an injective path) – YuiTo Cheng May 03 '19 at 04:58