For each positive integer n, let E(n) be n-dimensional Euclidean space. An answer to my MATHSTACKEXCHANGE question No. 2191335 informed me that Sierpinski-in an article in Volume 4(1923) of Fundamenta Mathematicae- had constructed an infinite closed and connected subset C of E(3) which is the countable union of (more than one) pairwise disjoint non-empty closed subsets of E(3). Unfortunately, I have not been able to obtain access to this article......My question is: Is there a theorem stating that no infinite closed and connected subset of E(2)-even if unbounded-can be the countable union of (more than one) pairwise disjoint non-empty closed subsets of E(2)?
This issue is discussed in Kuratowski's book "Topology", p. 175, which you should be able to find in a library (even online, if you know where to look). He discusses Sierpinski's example (the original is here) in detail. (But an embedding into the 3-space is not explicit.) Kuratowski also mentions (without detail) an example by Mazurkiewicz, in "Sur les continus plans non bornes", Fundamenta Math., Vol. 5, 1924, of a closed connected planar set which is a countable union of pairwise disjoint closed (but non-connected) nonempty subsets. (Sierpinski's example had a decomposition into closed connected subsets.) Moreover, Mazurkiewicz shows in the same paper that no closed connected planar set is a countable union of pairwise disjoint closed connected nonempty subsets.