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Miklós Schweitzer
2020 Miklós Schweitzer
5
5
Part of
2020 Miklós Schweitzer
Problems
(1)
equivalence on nowhere dense compact subset of the real plane
Source: Miklos Schweitzer 2020, Problem 5
12/1/2020
Prove that for a nowhere dense, compact set
K
⊂
R
2
K\subset \mathbb{R}^2
K
⊂
R
2
the following are equivalent:(i)
K
=
⋃
i
=
1
∞
K
n
K=\bigcup_{i=1}^{\infty}K_n
K
=
⋃
i
=
1
∞
K
n
where
K
n
K_n
K
n
is a compact set with connected complement for all
n
n
n
.(ii)
K
K
K
does not have a nonempty closed subset
S
⊆
K
S\subseteq K
S
⊆
K
such that any neighborhood of any point in
S
S
S
contains a connected component of
R
2
∖
S
\mathbb{R}^2 \setminus S
R
2
∖
S
.
real analysis