{{Short description|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 {{tmath|\R^n,}} {{math|''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 {{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.

== Construction ==

The following construction of the pseudo-arc follows {{harvtxt|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 <math>\mathcal{C}=\{C_1,C_2,\ldots,C_n\}</math> in a metric space such that <math>C_i\cap C_j\ne\emptyset</math> if and only if <math>|i-j|\le1.</math> 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 <math>\mathcal{C}</math> and <math>\mathcal{D}</math> be chains such that

:# each link of <math>\mathcal{D}</math> is a subset of a link of <math>\mathcal{C}</math>, and :# for any indices {{math|''i'', ''j'', ''m'', ''n''}} with <math>D_i\cap C_m\ne\emptyset</math>, <math>D_j\cap C_n\ne\emptyset</math>, and <math>m<n-2</math>, there exist indices <math>k</math> and <math>\ell</math> with <math>i<k<\ell<j</math> (or <math>i>k>\ell>j</math>) and <math>D_k\subseteq C_{n-1}</math> and <math>D_\ell\subseteq C_{m+1}.</math>

:Then <math>\mathcal{D}</math> is '''crooked''' in <math>\mathcal{C}.</math>

=== Pseudo-arc ===

For any collection {{mvar|C}} of sets, let {{mvar|C*}} denote the union of all of the elements of {{mvar|C}}. That is, let :<math>C^*=\bigcup_{S\in C}S.</math>

The ''pseudo-arc'' is defined as follows:

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

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

:Let ::<math>P=\bigcap_{i\in\mathbb{N}}\left(\mathcal{C}^i\right)^{*}.</math> :Then {{mvar|P}} is a '''pseudo-arc'''.

==Notes== {{notelist|notes=}}

== References == {{refbegin|}} * {{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}} * {{citation | last=Nadler | first1=Sam B. Jr. | title=Continuum theory. An introduction | series=Monographs and Textbooks in Pure and Applied Mathematics | volume=158 | publisher=Marcel Dekker, Inc., New York | date=1992 | isbn=0-8247-8659-9}} * {{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}} {{refend}}

Category:Continuum theory