Problem 910

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claims/: The 1 claim page of Problem 910, one per claimant's result; the problem's standing derives from them.

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Statement. Does every connected set in $\mathbb{R}^n$ contain a connected subset which is not a point and not homeomorphic to the original set?

If $n\geq 2$ does every connected set in $\mathbb{R}^n$ contain more than $2^{\aleph_0}$ many connected subsets?

Statement (corrected). Does every connected set in $\mathbb{R}^n$ 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 $n\geq 2$ does every connected set in $\mathbb{R}^n$ of dimension greater than one contain more than $2^{\aleph_0}$ many connected subsets?

Notes. 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 $2^c$ connected subsets", and [Er82e], p. 77, whether "every connected set (in a Euclidean space) of dimension greater than one contains more than $c=2^{\aleph_0}$ 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 $n\ge2$. 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 $2^c$ form of [Er44] is a variant.

Status. 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 [[problems/analysis/E0910/claims/1958_01_01_rudin|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.

Source. erdosproblems.com/910, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #910, https://www.erdosproblems.com/910.

References.

• [Ru58] Rudin, M. E., A connected subset of the plane. Fund. Math. 46 (1958), 15-24.
• [Er82e] Erdős, P., Some of my favourite problems which recently have been solved. Proceedings of the International Mathematical Conference (Singapore, 1981), North-Holland Math. Stud. 74 (1982), 59--79. Chapter V, §4, printed p. 77. Library home: erdos_1982_my_favourite_problems_which_recently_have.
• [Er44] Erdős, P., Some remarks on connected sets. Bull. Amer. Math. Soc. 50 (1944), 442--446, users.renyi.hu/~p_erdos/1944-06.pdf. The convention on points, printed pp. 443 and 445; the non-homeomorphism question and the arc remark, printed p. 445; the dimension question, printed p. 446. Not a site source key.

Formalization. None recorded.

Current assessment

Rudin’s theorem gives, under CH, a nondegenerate connected planar set with at most $\mathfrak c$ connected subsets, and its construction has not been reconstructed. Nothing on this page removes CH from Rudin’s theorem or settles either question of the corrected Statement in full.

The source statements below are recorded at the stated scopes; complete source proofs remain uncompiled. See Rudin's source record for the CH theorem and its distinct historical target.

Primary sources: Rudin 1958, Erdős 1944, and Erdős 1982; the physical and PDF page numbers on this page index these files.

No separate wider status-search date or scope is recorded on this page.

Progress

Rudin's A connected subset of the plane, Fundamenta Mathematicae 46 (1958), 15–24, explicitly assumes the continuum hypothesis (CH). Her theorem (printed p.15 / physical p.1) gives a nondegenerate connected planar set $M$ such that every nondegenerate connected subset $N\subseteq M$ has $M\setminus N$ at most countable. The full construction has not been reconstructed or independently reviewed here.

This property gives the following conditional consequence for the second question as the site words it. Write $\mathfrak c=2^{\aleph_0}$. Since $M\subseteq\mathbb R^2$, there are at most $\mathfrak c^{\aleph_0}=\mathfrak c$ at-most-countable subsets of $M$. The map $N\mapsto M\setminus N$ is injective, so there are at most $\mathfrak c$ nondegenerate connected subsets of $M$. Singletons add at most $\mathfrak c$ more subsets, and allowing the empty set does not change the bound. Under CH, therefore, the site's wording of the second question fails already in ambient dimension two.

The first clause requires a separate transfer. Rudin's direct citation is to Erdős's Some remarks on connected sets, Bull. Amer. Math. Soc. 50 (1944), printed p.443 / physical p.2: the conjecture that every nondegenerate connected $C$ has a nondegenerate connected subset $C'$ with $|C\setminus C'|=\mathfrak c$. That is different from the non-homeomorphism question on printed p.445 / physical p.4. The countable-complement theorem by itself does not show that all nondegenerate connected subsets of $M$ are homeomorphic to $M$. The first-clause implication remains unresolved on the checked materials.

The 1944 cardinality question is a variant. The 1944 paper, printed p.446 / physical p.5, asks for $2^{\mathfrak c}$ connected subsets when the connected set itself has dimension greater than one. The corrected second question keeps that hypothesis and asks, with Erdős 1982, for more than $\mathfrak c$ subsets, a weaker conclusion.

Erdős's Some of my favourite problems which recently have been solved (1982), §4, printed p.77 / physical p.19, lists the non-homeomorphism question and a more-than-$\mathfrak c$ question with intrinsic dimension greater than one, the corrected second question. Erdős attributes counterexamples to Rudin under CH. This is a historical attribution: the homeomorphism transfer it needs for the first question is unchecked. Physical p.18 is printed p.76 and concerns the different measurable-differences problem.

Rudin's optimality remark (printed p.15) says every nondegenerate connected set has a nondegenerate connected subset with infinite complement. It cites Erdős 1944 p.443; that external proof is not independently reviewed here. Rudin's printed p.24 / physical p.6 is §4, proving connectedness and the countable-complement property of her main construction, not the separate optimality remark.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• rudin_1958_connected_subset_plane