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equivalence on nowhere dense compact subset of the real plane

Source: Miklos Schweitzer 2020, Problem 5

December 1, 2020
real analysis

Problem Statement

Prove that for a nowhere dense, compact set K⊂R2K\subset \mathbb{R}^2 the following are equivalent:
(i) K=⋃i=1∞KnK=\bigcup_{i=1}^{\infty}K_n where KnK_n is a compact set with connected complement for all nn.
(ii) KK does not have a nonempty closed subset S⊆KS\subseteq K such that any neighborhood of any point in SS contains a connected component of R2∖S\mathbb{R}^2 \setminus S.
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