pseudo-arc

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.

The following construction of the pseudo-arc follows {{harvtxt|Lewis|1999}}.

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 $\backslash mathcal\{C\}=\backslash \{C\_1,C\_2,\backslash ldots,C\_n\backslash \}$ in a metric space such that $C\_i\backslash cap\; C\_j\backslash ne\backslash emptyset$ if and only if $|i-j|\backslash le1.$ 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 $\backslash mathcal\{C\}$ and $\backslash mathcal\{D\}$ be chains such that

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

:# for any indices {{math|i, j, m, n}} with $D\_i\backslash cap\; C\_m\backslash ne\backslash emptyset$, $D\_j\backslash cap\; C\_n\backslash ne\backslash emptyset$, and

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

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

:$C^*=\backslash bigcup\_\{S\backslash in\; C\}S.$

The pseudo-arc is defined as follows:

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

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

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

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

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

:Let

::$P=\backslash bigcap\_\{i\backslash in\backslash mathbb\{N\}\}\backslash left(\backslash mathcal\{C\}^i\backslash right)^\{*\}.$

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

{{notelist|notes=}}

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{{refend}}

Category:Continuum theory