Rudin 1958 connected subset plane

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evidence/: Retains the bounded independent review of the Rudin theorem's scope and of the separation of the Erdős 1944 and 1982 questions.

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Rudin, M. E., A connected subset of the plane. Fund. Math. 46 (1958), 15--24. https://doi.org/10.4064/fm-46-1-15-24.

Rudin constructs, assuming the continuum hypothesis, a non-degenerate connected subset M of the plane with the property that for every non-degenerate connected subset N of M the difference M - N is at most countable (Theorem, p. 15). This refutes under CH the complement-cardinality conjecture in Erdős 1944 p.443. She also states an optimality remark on p.15: every non-degenerate connected set M contains a non-degenerate connected subset N with M - N infinite. This remark cites Erdős 1944 p.443; its external proof is not independently reviewed here. Section 4 on printed p.24 proves the main construction’s connectedness and countable-complement property.

The construction proceeds in two stages: first an indecomposable continuum I in the plane is built as the intersection of nested chains of 2-cells of cyclically permuted types (a, b, c) with diameters below 1/n (Sections 1.1-1.3), and its sections, blocks, separating sets and composants are analyzed (1.4-1.5, including that every proper non-degenerate subcontinuum of I is an arc); then M is extracted from I by transfinite induction, meeting each composant of I in at most one point.

For Problem 910, the countable-complement property bounds the number of nondegenerate connected subsets by the at-most-countable subsets of a planar set, hence by c = 2^{aleph_0}; adding singletons and the possible empty set still gives at most c. Thus the current ambient-dimension second clause has a CH-conditional counterexample. The theorem’s direct target on Erdős 1944 p.443 differs from the non-homeomorphism question on p.445. A transfer showing that every nondegenerate connected subset is homeomorphic to M has not been established here. Erdős’s 1982 report, Some of my favourite problems which recently have been solved, printed p.77 / physical p.19, attributes counterexamples to Rudin under CH; that historical statement does not supply the missing theorem-level transfer. No unconditional conclusion is inferred from Rudin’s CH theorem. The 1944 cardinality question on p.446 uses intrinsic dimension greater than one and 2^c subsets; this differs from current ambient n >= 2 and more than c. No full construction proof or intrinsic-dimension transfer is reviewed here.

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Bears on. #910

Results to transcribe.

• Theorem (p. 15): If the continuum hypothesis holds, there is a non-degenerate connected subset M of the plane such that for every non-degenerate connected subset N of M the set M - N is at most countable. The direct target is the complement-cardinality conjecture of Erdős 1944 p.443; the current Problem 910 second-clause transfer is conditional on CH.
• Remark (p. 15): The theorem cannot be strengthened: every non-degenerate connected set M contains a non-degenerate connected subset N with M - N infinite.
• Section 1.5(b): For the constructed indecomposable continuum I, the intersection of a composant C with a section U is the union of countably many of the components of U, each an arc segment (an arc minus its end points); the proof of 1.5(b) (from p. 17) shows along the way that every proper non-degenerate subcontinuum of I is an arc.

Conventions and source boundary. Erdős 1944 p. 445 excludes points from the connected sets in the non-homeomorphism question. For that intended question, both original and requested sets are taken to have at least two points. No theorem-level homeomorphism or intrinsic-dimension transfer is claimed. Printed pp. 15 and 24 were locally re-read; the full construction remains outside this proof scope.

Review record. The formulation review of 6 September 2026 checks the scope of Rudin's theorem on printed p. 15. No record reviews Rudin's construction.