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Let E be a finite dimensional Euclidean space. A well known theorem of Sierpinski states that if C is an infinite compact-i.e. closed and bounded-connected subset of E, then C cannot be the countable union of pairwise disjoint closed subsets of E. I have also seen a theorem of Hausdorff which implies that the same conclusion continues to hold if C is any infinite closed connected subset of E-not necessarily compact-provided C is locally connected. My question is: Does Hausdorff's theorem continue to hold, if one drops the requirement that C should be locally connected?..........Some of the statements I have read in the literature seem to suggest that the answer is "Yes" whenever the infinite connected subset C of E is locally compact. This will certainly be the case if E is a finite dimensional Euclidean space and C is a closed subset of E. But I am not really clear about the whole situation.

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Sierpinsky himself constructed an example of a closed connected subset of $R^3$ which is a countable union of pairwise disjoint closed nonempty subsets.

W. Sierpinski, Sur guelgues proprietes topologiques du plan, Fund. Math. 4 (1923), 1-6.

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  • $\begingroup$ This is the answer I was looking for. It shows that additional conditions on C besides being closed in E are needed, in order for Sierpinski's conclusion to follow. $\endgroup$ – Garabed Gulbenkian Mar 19 '17 at 19:47

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