---

Claim. Assume the continuum hypothesis. Then there is a nondegenerate connected set $M\subset\mathbb{R}^2$ such that for every nondegenerate connected $N\subseteq M$ the difference $M\setminus N$ is at most countable. This is the theorem of Rudin's paper, whose digest is the source card; its construction is not reconstructed in this corpus. The site credits this set, under the continuum hypothesis, with a negative answer to both questions of Problem 910, as does Erdős's 1982 retrospective, which attributes counterexamples to Rudin under that hypothesis.

Hypothesis. The continuum hypothesis, $2^{\aleph_0}=\aleph_1$, which the paper assumes and which is independent of the usual axioms of set theory; the claim gives no unconditional answer.

Scope. The scope is judged against the problem's corrected Statement. For the second question, Erdős's 1982 attribution and the site's credit read $M$ as a counterexample; under the site's ambient wording this follows from the theorem under the continuum hypothesis: the nondegenerate connected subsets of $M$ are determined by their at-most-countable complements, of which a planar set has at most $\mathfrak c=2^{\aleph_0}$, so $M$, and its copy in $\mathbb{R}^n$ for any $n\geq2$, has at most $\mathfrak c$ connected subsets. For the first question the paper's direct target is the conjecture of Erdős's 1944 note that every nondegenerate connected set has a nondegenerate connected subset with complement of cardinality $\mathfrak c$; the paper neither states nor proves that every nondegenerate connected subset of $M$ is homeomorphic to $M$, and this corpus has not found that step, so the credit for the first question rests on Erdős's 1982 attribution and the curator's.

Acceptance. The result is refereed: M. E. Rudin, A connected subset of the plane, Fundamenta Mathematicae 46 (1958), 15–24. It is reviewed in the sense of a documented independent acceptance: Thomas Bloom, the curator of erdosproblems.com, marks Problem 910 disproved and credits this paper, conditional on the continuum hypothesis. No Lean proof is recorded. Being conditional, the claim settles no standing of the problem by itself.

The page is dated to the year of publication; the scan prints no fuller date for the article.