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Are any examples known of infinite metric spaces that are connected, locally connected, and sigma compact but not arc-wise connected?

Garabed Gulbenkian
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  • Have you checked that the examples in my other answer (all 3 of them) are all non-$\sigma$-compact? – Henno Brandsma Feb 28 '16 at 17:50
  • I mean the answer http://math.stackexchange.com/a/1669089/4280, for reference. – Henno Brandsma Feb 28 '16 at 18:28
  • I have been looking at the Knaster and Kuratowski article. His set X is a subset of the Sierpinski curve and I am having a lot of trouble trying to work out whether X is or is not sigma-compact. I do not have access to a math library and can only read articles which I can download from my computer. I am not sure whether I can get Moore's article this way but I am trying. I don't see a way of getting hold of the topology book. But I really appreciate all the help you have given me. My own efforts to answer the question were getting nowhere. – Garabed Gulbenkian Feb 28 '16 at 22:00
  • I don't think that K and K's set X is sigma-compact. Let C be any compact subset of X. If S is any non-degenerate connected subset of C and if S* is the closure of S, then S* is closed and uncountable. Hence S* contains a perfect subset and X is not allowed to contain perfect subsets. Consequently C is totally disconnected and therefore zero-dimensional. But if X is sigma-compact, then X is zero-dimensional and so cannot be connected. Contradiction! – Garabed Gulbenkian Feb 29 '16 at 21:01
  • That last argument looks right (using the countable closed sum theorem from dimension theory). That kills one potential counterexample. – Henno Brandsma Feb 29 '16 at 21:10
  • I was able to access the article of R.L, Moore. His set M is a countable union of closed subsets of the Euclidean plane. Since the Euclidean plane is sigma-compact, this makes M sigma-compact. So here is an example of the kind of metric space I have been looking for. Moore does not mention sigma-compactness as a further noteworthy property of his set M. But maybe sigma-compactness was not of much interest to topologists at that time. – Garabed Gulbenkian Mar 01 '16 at 20:40
  • his observation is equivalent and the term sigma-compact might not even have been coined yet. – Henno Brandsma Mar 01 '16 at 20:42
  • Could you perhaps answer your own question, having found the answer you're looking for? I would like to see a well-explained answer to this myself please! – Morgan Rogers Aug 07 '16 at 15:39

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