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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 Modèle:Tmath $n \geq 2$ , are homeomorphic to the pseudo-arc.

History

In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane Modèle:Tmath must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in Modèle:Tmath 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.^{[note 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.^{[note 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 Modèle:Harvtxt.

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}, \dots, C_{n}}

in a metric space such that

C_{i} \cap C_{j} \neq ∅︀

if and only if

| 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

and

𝒟

be chains such that

each link of $𝒟$ is a subset of a link of $𝒞$ , and
for any indices $i, j, m, n$ with $D_{i} \cap C_{m} \neq ∅︀$ , $D_{j} \cap C_{n} \neq ∅︀$ , and $m < n - 2$ , there exist indices $k$ and $ℓ$ with $i < k < ℓ < j$ (or $i > k > ℓ > j$ ) and $D_{k} \subseteq C_{n - 1}$ and $D_{ℓ} \subseteq C_{m + 1} .$

Then

𝒟

is crooked in

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 \in C} S .

The pseudo-arc is defined as follows:

Let

p, q

be distinct points in the plane and

{𝒞^{i}}_{i \in ℕ}

be a sequence of chains in the plane such that for each

i

,

the first link of $𝒞^{i}$ contains $p$ and the last link contains $q$ ,
the chain $𝒞^{i}$ is a $1 / 2^{i}$ -chain,
the closure of each link of $𝒞^{i + 1}$ is a subset of some link of $𝒞^{i}$ , and
the chain $𝒞^{i + 1}$ is crooked in $𝒞^{i}$ .

Let

P = ⋂_{i \in ℕ} {(𝒞^{i})}^{*} .

Then

P

is a pseudo-arc.

Notes

Notelist

References

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↑ Modèle:Harvtxt later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.
↑ The history of the discovery of the pseudo-arc is described in Modèle:Harvtxt, pp. 228–229.

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[1] Modèle:Harvtxt 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 Modèle:Harvtxt, pp. 228–229.

[note 1]

[note 2]