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 R n , {\displaystyle \mathbb {R} ^{n},} 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 R 2 {\displaystyle \mathbb {R} ^{2}} must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R 2 {\displaystyle \mathbb {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 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. 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 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 C = { C 1 , C 2 , … , C n } {\displaystyle {\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{n}\}} in a metric space such that C i ∩ C j ≠ ∅ {\displaystyle C_{i}\cap C_{j}\neq \emptyset } if and only if | i − j | ≤ 1. {\displaystyle |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:
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