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Does every connected set in contain a connected subset which is not a point and not homeomorphic to the original set?
If does every connected set in contain more than many connected subsets?
Does every connected set in contain a connected subset which is not a point and not homeomorphic to the original set? (Points and the empty set do not count as connected sets.)
If does every connected set in of dimension greater than one contain more than many connected subsets?
Source: erdosproblems.com/910
No claim settles this problem.
The site labels the problem DISPROVED on the strength of Rudin's 1958 construction [Ru58]: its commentary credits the construction, under the continuum hypothesis, with a negative answer to both questions, a result recorded as an accepted conditional claim. The page departs from the DISPROVED label, as the Notes explain: the homeomorphism step that the credit needs for the first question is not in the paper and is unchecked here. A conditional claim derives nothing in any case, so the standing is open.
Both questions are defective as the site words them: the first does not exclude points or the empty set, and the second reads Erdős's dimension hypothesis as the ambient dimension. The site labels the problem DISPROVED and its commentary says that "the answer to both is in fact no, as shown by Rudin [Ru58] (conditional on the continuum hypothesis)"; its thread has no comments. The corrected Statement follows Erdős's own words, which the curator's credit presupposes for the first question: [Er44], printed p. 445, asks "Is it true that every connected set contains a connected subset not homeomorphic to it? (Points do not count as connected sets.)", p. 443 states the same convention, and [Er82e], p. 77, asks for a subset "which is not a point"; the change inserts "(Points and the empty set do not count as connected sets.)". For the second question [Er44], printed p. 446, asks whether "every connected set of dimension greater than 1 contains connected subsets", and [Er82e], p. 77, whether "every connected set (in a Euclidean space) of dimension greater than one contains more than connected subsets"; the hypothesis is on the set's own dimension, and the change inserts "of dimension greater than one". The site's wording reads that phrase as the ambient dimension . The curator's verdict follows [Er82e], where Erdős writes that "Mary Ellen Rudin using the continuum hypothesis found the required counter examples" to both questions. For the first question Rudin's theorem gives countable complements, and the paper neither states nor proves that every nondegenerate connected subset of her set is homeomorphic to the set; that step is asserted by [Er82e] and the site and has not been found here. The first defect is already in [Er82e], which excludes points for the subset only; the second is the site's. The form of [Er44] is a variant.