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Is there any example of a path component that is neither open nor closed?

Usually the topologist’s sine curve is used as an example, however that is not enough to answer my question because its path components are still either open or closed. I couldn’t find anything better so far.

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As always, any comment or answer is welcome and let me know if I can explain myself clearer!

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Consider a subset of $\mathbb{R}^2$ made up of two disjoint curves, where each curve accumulates along part of the other like a topologist's sine curve. This space then has two path components (the two curves) which are neither open nor closed (since each has accumulation points inin the other).

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  • $\begingroup$ Thank you for the answer! However I can not see why the path components of your space are not open. $\endgroup$ Sep 13, 2022 at 17:50
  • $\begingroup$ Their complements are not closed. $\endgroup$ Sep 13, 2022 at 18:55
  • $\begingroup$ Oh right! Thank you very much! $\endgroup$ Sep 13, 2022 at 23:10

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