# A question about how far a well known theorem of Sierpinski can be strengthened

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.

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