- Source: Pseudo-arc
In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. Bing proved that, in a certain well-defined sense, most continua in
R
n
,
{\displaystyle \mathbb {R} ^{n},}
n ≥ 2, are homeomorphic to the pseudo-arc.
History
In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane
R
2
{\displaystyle \mathbb {R} ^{2}}
must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in
R
2
{\displaystyle \mathbb {R} ^{2}}
that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster discovered the first example of a hereditarily indecomposable continuum K, later named the pseudo-arc, giving a negative answer to a Mazurkiewicz question. In 1948, R. H. Bing proved that Knaster's continuum is homogeneous, i.e. for any two of its points there is a homeomorphism taking one to the other. Yet also in 1948, Edwin Moise showed that Knaster's continuum is homeomorphic to each of its non-degenerate subcontinua. Due to its resemblance to the fundamental property of the arc, namely, being homeomorphic to all its nondegenerate subcontinua, Moise called his example M a pseudo-arc. Bing's construction is a modification of Moise's construction of M, which he had first heard described in a lecture. In 1951, Bing proved that all hereditarily indecomposable arc-like continua are homeomorphic — this implies that Knaster's K, Moise's M, and Bing's B are all homeomorphic. Bing also proved that the pseudo-arc is typical among the continua in a Euclidean space of dimension at least 2 or an infinite-dimensional separable Hilbert space. Bing and F. Burton Jones constructed a decomposable planar continuum that admits an open map onto the circle, with each point preimage homeomorphic to the pseudo-arc, called the circle of pseudo-arcs. Bing and Jones also showed that it is homogeneous. In 2016 Logan Hoehn and Lex Oversteegen classified all planar homogeneous continua, up to a homeomorphism, as the circle, pseudo-arc and circle of pseudo-arcs. In 2019 Hoehn and Oversteegen showed that the pseudo-arc is topologically the only, other than the arc, hereditarily equivalent planar continuum, thus providing a complete solution to the planar case of Mazurkiewicz's problem from 1921.
Construction
The following construction of the pseudo-arc follows Lewis (1999).
= Chains
=At the heart of the definition of the pseudo-arc is the concept of a chain, which is defined as follows:
A chain is a finite collection of open sets
C
=
{
C
1
,
C
2
,
…
,
C
n
}
{\displaystyle {\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{n}\}}
in a metric space such that
C
i
∩
C
j
≠
∅
{\displaystyle C_{i}\cap C_{j}\neq \emptyset }
if and only if
|
i
−
j
|
≤
1.
{\displaystyle |i-j|\leq 1.}
The elements of a chain are called its links, and a chain is called an ε-chain if each of its links has diameter less than ε.
While being the simplest of the type of spaces listed above, the pseudo-arc is actually very complex. The concept of a chain being crooked (defined below) is what endows the pseudo-arc with its complexity. Informally, it requires a chain to follow a certain recursive zig-zag pattern in another chain. To 'move' from the m-th link of the larger chain to the n-th, the smaller chain must first move in a crooked manner from the m-th link to the (n − 1)-th link, then in a crooked manner to the (m + 1)-th link, and then finally to the n-th link.
More formally:
Let
C
{\displaystyle {\mathcal {C}}}
and
D
{\displaystyle {\mathcal {D}}}
be chains such that
each link of
D
{\displaystyle {\mathcal {D}}}
is a subset of a link of
C
{\displaystyle {\mathcal {C}}}
, and
for any indices i, j, m, n with
D
i
∩
C
m
≠
∅
{\displaystyle D_{i}\cap C_{m}\neq \emptyset }
,
D
j
∩
C
n
≠
∅
{\displaystyle D_{j}\cap C_{n}\neq \emptyset }
, and
m
<
n
−
2
{\displaystyle m
, there exist indices
k
{\displaystyle k}
and
ℓ
{\displaystyle \ell }
with
i
<
k
<
ℓ
<
j
{\displaystyle i
(or
i
>
k
>
ℓ
>
j
{\displaystyle i>k>\ell >j}
) and
D
k
⊆
C
n
−
1
{\displaystyle D_{k}\subseteq C_{n-1}}
and
D
ℓ
⊆
C
m
+
1
.
{\displaystyle D_{\ell }\subseteq C_{m+1}.}
Then
D
{\displaystyle {\mathcal {D}}}
is crooked in
C
.
{\displaystyle {\mathcal {C}}.}
= Pseudo-arc
=For any collection C of sets, let C* denote the union of all of the elements of C. That is, let
C
∗
=
⋃
S
∈
C
S
.
{\displaystyle C^{*}=\bigcup _{S\in C}S.}
The pseudo-arc is defined as follows:
Let p, q be distinct points in the plane and
{
C
i
}
i
∈
N
{\displaystyle \left\{{\mathcal {C}}^{i}\right\}_{i\in \mathbb {N} }}
be a sequence of chains in the plane such that for each i,
the first link of
C
i
{\displaystyle {\mathcal {C}}^{i}}
contains p and the last link contains q,
the chain
C
i
{\displaystyle {\mathcal {C}}^{i}}
is a
1
/
2
i
{\displaystyle 1/2^{i}}
-chain,
the closure of each link of
C
i
+
1
{\displaystyle {\mathcal {C}}^{i+1}}
is a subset of some link of
C
i
{\displaystyle {\mathcal {C}}^{i}}
, and
the chain
C
i
+
1
{\displaystyle {\mathcal {C}}^{i+1}}
is crooked in
C
i
{\displaystyle {\mathcal {C}}^{i}}
.
Let
P
=
⋂
i
∈
N
(
C
i
)
∗
.
{\displaystyle P=\bigcap _{i\in \mathbb {N} }\left({\mathcal {C}}^{i}\right)^{*}.}
Then P is a pseudo-arc.
Notes
References
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