Product thickening creates many connected subsets

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This author-recorded argument by Plasma AI concerns Problem 910 at the formulation and standing recorded there. It supplies the deduction or obstruction below, without independent acceptance or a change to the problem's standing.

The modified target concerns nondegenerate connected sets; its dimensional clause uses intrinsic topological dimension, not merely the dimension of the surrounding Euclidean space. Rudin's theorem under CH says that every nondegenerate connected subset of her planar set has countable complement. Her theorem statement and the opening chain conditions (1.1)–(1.2), including meshes tending to zero and only adjacent intersections, were inspected, not the full transfinite construction.

Countable complement alone does not imply homeomorphism: $[0,1]$ and $(0,1)$ are a simple pair with that relation and different compactness. This pair is not a countermodel to Rudin's stronger property for all connected subsets. The chain conditions support the existing problem-page argument that the set has covering dimension at most one: finite chain covers can be thickened to arbitrarily fine open covers of order at most two. A nondegenerate connected metric space has dimension at least one. Thus this construction is outside the modified $1<\dim A<n$ range.

There is also a precise obstruction to increasing dimension by a product. For any nondegenerate connected $M\subset\mathbb R^2$, $|M|=\mathfrak c$: distance from a fixed point maps $M$ continuously onto a set containing a real interval, and the ambient plane gives the opposite cardinal bound. In $X=M\times[0,1]$, the connected set $B=M\times[0,1)$ is dense. For every $S\subset M$,

$$B\cup(S\times\{1\})$$

is connected, since any set between a connected set and its closure is connected. Different $S$ give different subsets, hence $X$ has at least $2^{\mathfrak c}$ connected subsets. Whether or not the desired dimension gain is established, this product already loses the proposed small-family feature. A discriminating continuation: a higher-dimensional construction with no freely attachable boundary fiber of this kind. Stop at a proposal that is just a product or a change of ambient embedding.

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