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I have added a clarification excluding singletons from the question statement, because the reference provided excludes singletons as well (see page 100, point 17)

edit approved Sep 2, 2018 at 21:57

Stefano Gogioso

Path connected without bounded path connected subset?

Question: Is there a path connected subset of $\mathbb R^2$, without any bounded path connected subset (aside from singletons)?

Motivation: If we replace "path connected" by "connected", then the answer is positive. It was proven by Stefan Mazurkiewicz, see the original article.

gn.general-topology path-connected

asked Sep 2, 2018 at 14:37

Portland

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