Pseudo-arc

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Type of topological continuum

title: "Pseudo-arc" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["continuum-theory"] description: "Type of topological continuum" topic_path: "general/continuum-theory" source: "https://en.wikipedia.org/wiki/Pseudo-arc" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Type of topological continuum ::

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 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 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in 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 .

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 \mathcal{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|\le1. 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 \mathcal{C} and \mathcal{D} be chains such that

:# each link of \mathcal{D} is a subset of a link of \mathcal{C}, and :# for any indices i, j, m, n with D_i\cap C_m\ne\emptyset, D_j\cap C_n\ne\emptyset, and m, there exist indices k and \ell with i (or ik\ellj) and D_k\subseteq C_{n-1} and D_\ell\subseteq C_{m+1}.

:Then \mathcal{D} is crooked in \mathcal{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^*=\bigcup_{S\in C}S.

The pseudo-arc is defined as follows:

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

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

:Let ::P=\bigcap_{i\in\mathbb{N}}\left(\mathcal{C}^i\right)^{*}. :Then P is a pseudo-arc.

Notes

References

  • {{citation | last1=Bing | first1=R.H. | author-link1=R. H. Bing | title=A homogeneous indecomposable plane continuum | journal=Duke Mathematical Journal | volume=15 | issue=3 | date=1948 | pages=729–742 | doi=10.1215/S0012-7094-48-01563-4}}
  • {{citation | last1=Bing | first1=R.H. | author-link1=R. H. Bing | title=Concerning hereditarily indecomposable continua | journal=Pacific Journal of Mathematics | volume=1 | date=1951 | pages=43–51 | doi=10.2140/pjm.1951.1.43 | doi-access=free}}
  • {{citation | last1=Bing | first1=R.H. | author-link1=R. H. Bing | last2=Jones | first2=F. Burton | title=Another homogeneous plane continuum | journal=Transactions of the American Mathematical Society | volume=90 | issue=1 | date=1959 | pages=171–192 | doi=10.1090/S0002-9947-1959-0100823-3 | doi-access=free}}
  • {{citation | last1=Henderson | first1=George W. | title=Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc | journal=Annals of Mathematics | series=2nd series | volume=72 | issue=3 | date=1960 | pages=421–428 | doi=10.2307/1970224}}
  • {{citation | last1=Hoehn | first1=Logan C. | last2=Oversteegen | first2=Lex G. | title=A complete classification of homogeneous plane continua | journal=Acta Mathematica | volume=216 | date=2016 | issue=2 | pages=177–216 | doi=10.1007/s11511-016-0138-0 | doi-access=free| arxiv=1409.6324
  • {{citation | last1=Hoehn | first1=Logan C. | last2=Oversteegen | first2=Lex G. | title=A complete classification of hereditarily equivalent plane continua | journal=Advances in Mathematics | volume=368 | date=2020 | article-number=107131 | arxiv=1812.08846 | doi=10.1016/j.aim.2020.107131 | doi-access=free}}
  • {{citation | last1=Irwin | first1=Trevor | last2=Solecki | first2=Sławomir | title=Projective Fraïssé limits and the pseudo-arc | journal=Transactions of the American Mathematical Society | volume=358 | issue=7 | date=2006 | pages=3077–3096 | doi=10.1090/S0002-9947-06-03928-6 | doi-access=free}}
  • {{citation | last1=Kawamura | first1=Kazuhiro | title=On a conjecture of Wood | journal=Glasgow Mathematical Journal | volume=47 | issue=1 | date=2005 | pages=1–5 | doi=10.1017/S0017089504002186 | doi-access=free}}
  • {{citation | last1=Knaster | first1=Bronisław | author-link1=Bronisław Knaster | title=Un continu dont tout sous-continu est indécomposable | journal=Fundamenta Mathematicae | volume=3 | date=1922 | pages=247–286 | doi=10.4064/fm-3-1-247-286 | doi-access=free}}
  • {{citation | last1=Lewis | first1=Wayne | title=The Pseudo-Arc | journal=Boletín de la Sociedad Matemática Mexicana | volume=5 | issue=1 | date=1999 | pages=25–77}}
  • {{citation | last1=Lewis | first1=Wayne | last2=Minc | first2=Piotr | title=Drawing the pseudo-arc | url=https://www.math.uh.edu/~hjm/restricted/pdf36(3)/16lewis.pdf | journal=Houston Journal of Mathematics | volume=36 | date=2010 | pages=905–934}}
  • {{citation | last1=Moise | first1=Edwin | author-link1=Edwin Moise | title=An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua | journal=Transactions of the American Mathematical Society | volume=63 | issue=3 | date=1948 | pages=581–594 | doi=10.1090/S0002-9947-1948-0025733-4 | doi-access=free}}
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  • {{citation | last1=Rambla | first1=Fernando | title=A counterexample to Wood's conjecture | journal=Journal of Mathematical Analysis and Applications | volume=317 | issue=2 | date=2006 | pages=659–667 | doi=10.1016/j.jmaa.2005.07.064 | doi-access=free}}
  • {{citation | last1=Rempe-Gillen | first1=Lasse | title=Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture | arxiv=1610.06278 | date=2016}}

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