Pseudo-arc explained

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. 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. 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 .

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

l{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|\le 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 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

l{C} and l{D} be chains such that

1. each link of

l{D} is a subset of a link of l{C} , and

1. for any indices i, j, m, and n with

D _{i\cap} C _{m\ne\emptyset} , D _{j\cap} C _{n\ne\emptyset} , 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

l{D} is crooked in l{C}.

Pseudo-arc

For any collection C of sets, let

C ^{*} denote the union of all of the elements of C. That is, letC ^{* = c up} _{S\in C} S.

The pseudo-arc is defined as follows:

Let p and q be distinct points in the plane and

\left\{ l{C} ^{i} \right\} _{i\inN} be a sequence of chains in the plane such that for each i,

1. the first link of

l{C} ^{i} contains p and the last link contains q,

1. the chain

l{C} ^{i} is a 1 / 2 ^{i} -chain,

1. the closure of each link of

l{C} ^{i+1} is a subset of some link of l{C} ^{i} , and

1. the chain

l{C} ^{i+1} is crooked in l{C} ^{i} .

Let

P = c ap _{i\inN} \left( l{C} ^{i\right)} ^{*} .

Then P is a pseudo-arc.