Pseudo-arc

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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 Rn, 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 R2 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R2 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.[lower-alpha 1] 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.[lower-alpha 2] 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 (Wayne 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 [math]\displaystyle{ \mathcal{C}=\{C_1,C_2,\ldots,C_n\} }[/math] in a metric space such that [math]\displaystyle{ C_i\cap C_j\ne\emptyset }[/math] if and only if [math]\displaystyle{ |i-j|\le1. }[/math] 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 mth link of the larger chain to the nth, the smaller chain must first move in a crooked manner from the mth link to the (n − 1)th link, then in a crooked manner to the (m + 1)th link, and then finally to the nth link.

More formally:

Let [math]\displaystyle{ \mathcal{C} }[/math] and [math]\displaystyle{ \mathcal{D} }[/math] be chains such that
  1. each link of [math]\displaystyle{ \mathcal{D} }[/math] is a subset of a link of [math]\displaystyle{ \mathcal{C} }[/math], and
  2. for any indices i, j, m, and n with [math]\displaystyle{ D_i\cap C_m\ne\emptyset }[/math], [math]\displaystyle{ D_j\cap C_n\ne\emptyset }[/math], and [math]\displaystyle{ m\lt n-2 }[/math], there exist indices [math]\displaystyle{ k }[/math] and [math]\displaystyle{ \ell }[/math] with [math]\displaystyle{ i\lt k\lt \ell\lt j }[/math] (or [math]\displaystyle{ i\gt k\gt \ell\gt j }[/math]) and [math]\displaystyle{ D_k\subseteq C_{n-1} }[/math] and [math]\displaystyle{ D_\ell\subseteq C_{m+1}. }[/math]
Then [math]\displaystyle{ \mathcal{D} }[/math] is crooked in [math]\displaystyle{ \mathcal{C}. }[/math]

Pseudo-arc

For any collection C of sets, let [math]\displaystyle{ C^{*} }[/math] denote the union of all of the elements of C. That is, let

[math]\displaystyle{ C^*=\bigcup_{S\in C}S. }[/math]

The pseudo-arc is defined as follows:

Let p and q be distinct points in the plane and [math]\displaystyle{ \left\{\mathcal{C}^{i}\right\}_{i\in\mathbb{N}} }[/math] be a sequence of chains in the plane such that for each i,
  1. the first link of [math]\displaystyle{ \mathcal{C}^i }[/math] contains p and the last link contains q,
  2. the chain [math]\displaystyle{ \mathcal{C}^i }[/math] is a [math]\displaystyle{ 1/2^i }[/math]-chain,
  3. the closure of each link of [math]\displaystyle{ \mathcal{C}^{i+1} }[/math] is a subset of some link of [math]\displaystyle{ \mathcal{C}^i }[/math], and
  4. the chain [math]\displaystyle{ \mathcal{C}^{i+1} }[/math] is crooked in [math]\displaystyle{ \mathcal{C}^i }[/math].
Let
[math]\displaystyle{ P=\bigcap_{i\in\mathbb{N}}\left(\mathcal{C}^i\right)^{*}. }[/math]
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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