Pseudo-arc

Template:Short description 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 Template:Tmath Template:Math, are homeomorphic to the pseudo-arc.

History

In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane Template:Tmath must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in Template: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 Template:Mvar, 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 Template:Mvar a pseudo-arc.Template:Efn Bing's construction is a modification of Moise's construction of Template:Mvar, 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 Template:Mvar, Moise's Template:Mvar, and Bing's Template:Mvar 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.Template:Efn 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 Template: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 $C_i\cap C_j\ne \mathrm{\varnothing }$ if and only if $|i-j|\le 1.$ The elements of a chain are called its links, and a chain is called an Template:Mvar-chain if each of its links has diameter less than Template:Mvar.

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 Template:Mvar-th link of the larger chain to the Template:Mvar-th, the smaller chain must first move in a crooked manner from the Template:Mvar-th link to the Template:Math-th link, then in a crooked manner to the Template:Math-th link, and then finally to the Template:Mvar-th link.

More formally:

Let $𝒞$ and $𝒟$ be chains such that

1. each link of $𝒟$ is a subset of a link of $𝒞$, and
2. for any indices Template:Math with $D_i\cap C_m\ne \mathrm{\varnothing }$, $D_j\cap C_n\ne \mathrm{\varnothing }$, and $m<n-2$, there exist indices $k$ and $\ell$ with $i<k<\ell <j$ (or $i>k>\ell >j$) and $D_k\subseteq C_{n-1}$ and $D_{\ell }\subseteq C_{m+1}.$

Then $𝒟$ is crooked in $𝒞.$

Pseudo-arc

For any collection Template:Mvar of sets, let Template:Mvar denote the union of all of the elements of Template:Mvar. That is, let

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

The pseudo-arc is defined as follows:

Let Template:Math be distinct points in the plane and ${\left\{{𝒞}^i\right\}}_{i\in \mathbb{N}}$ be a sequence of chains in the plane such that for each Template:Mvar,

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

Let

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

Then Template:Mvar is a pseudo-arc.

References

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