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Let $X$ be a path-connected topological space and $Y$ an open connected subset. Is $Y$ path-connected?

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Let $X$ be the topologist's sine curve, together with a path linking the two endpoints. Then $X$ is path-connected but deleting a point (other than the origin) yields a space that is connected but not path-connected.

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  • $\begingroup$ As an observation, note that the assumption of locally path-connectedness on $X$ would imply a positive result: since then you would be able to prove that a path-connected component of $Y$ would be both open. $\endgroup$ – Aloizio Macedo Aug 10 '15 at 7:23

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