pseudo-arc

{{Short description|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 {{tmath|\R^n,}} {{math|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 {{tmath|\R^2}} must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in {{tmath|\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 {{mvar|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 {{mvar|M}} a pseudo-arc.{{efn|{{harvtxt|Henderson|1960}} later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.}} Bing's construction is a modification of Moise's construction of {{mvar|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 {{mvar|K}}, Moise's {{mvar|M}}, and Bing's {{mvar|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.{{efn|The history of the discovery of the pseudo-arc is described in {{harvtxt|Nadler|1992}}, pp. 228–229.}} 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. A continuum is called "hereditarily equivalent" if it is homeomorphic to each of its non-degenerate sub-continua. In 2019 Hoehn and Oversteegen showed that the single point, the arc, and the pseudo-arc are topologically the only hereditarily equivalent planar continua, thus providing a complete solution to the planar case of Mazurkiewicz's problem from 1921.

Construction

The following construction of the pseudo-arc follows {{harvtxt|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 \mathcal{C}=\{C_1,C_2,\ldots,C_n\} in a metric space such that C_i\cap C_j\ne\emptyset if and only if |i-j|\le1. The elements of a chain are called its links, and a chain is called an {{mvar|ε}}-chain if each of its links has diameter less than {{mvar|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 {{mvar|m}}-th link of the larger chain to the {{mvar|n}}-th, the smaller chain must first move in a crooked manner from the {{mvar|m}}-th link to the {{math|(n − 1)}}-th link, then in a crooked manner to the {{math|(m + 1)}}-th link, and then finally to the {{mvar|n}}-th link.

More formally:

:Let \mathcal{C} and \mathcal{D} be chains such that

:# each link of \mathcal{D} is a subset of a link of \mathcal{C}, and

:# for any indices {{math|i, j, m, n}} with D_i\cap C_m\ne\emptyset, D_j\cap C_n\ne\emptyset, and m, there exist indices k and \ell with i (or i>k>\ell>j) and D_k\subseteq C_{n-1} and D_\ell\subseteq C_{m+1}.

:Then \mathcal{D} is crooked in \mathcal{C}.

= Pseudo-arc =

For any collection {{mvar|C}} of sets, let {{mvar|C*}} denote the union of all of the elements of {{mvar|C}}. That is, let

:C^*=\bigcup_{S\in C}S.

The pseudo-arc is defined as follows:

:Let {{math|p, q}} be distinct points in the plane and \left\{\mathcal{C}^{i}\right\}_{i\in\N} be a sequence of chains in the plane such that for each {{mvar|i}},

:#the first link of \mathcal{C}^i contains {{mvar|p}} and the last link contains {{mvar|q}},

:#the chain \mathcal{C}^i is a 1/2^i-chain,

:#the closure of each link of \mathcal{C}^{i+1} is a subset of some link of \mathcal{C}^i, and

:#the chain \mathcal{C}^{i+1} is crooked in \mathcal{C}^i.

:Let

::P=\bigcap_{i\in\mathbb{N}}\left(\mathcal{C}^i\right)^{*}.

:Then {{mvar|P}} is a pseudo-arc.

Notes

{{notelist|notes=}}

References

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{{refend}}

Category:Continuum theory