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Pseudo-arc

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Type of topological continuum

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
  1. each link of D {\displaystyle {\mathcal {D}}} is a subset of a link of C {\displaystyle {\mathcal {C}}} , and
  2. 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<n-2} , there exist indices k {\displaystyle k} and ℓ {\displaystyle \ell } with i < k < ℓ < j {\displaystyle i<k<\ell <j} (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,
  1. the first link of C i {\displaystyle {\mathcal {C}}^{i}} contains p and the last link contains q,
  2. the chain C i {\displaystyle {\mathcal {C}}^{i}} is a 1 / 2 i {\displaystyle 1/2^{i}} -chain,
  3. 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
  4. 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

  1. Henderson (1960) later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.
  2. The history of the discovery of the pseudo-arc is described in Nadler (1992), pp. 228–229.

References

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