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| In [[general topology]], the '''pseudo-arc''' is the simplest nondegenerate [[Hereditary property#In topology|hereditarily]] [[indecomposable continuum]]. The pseudo-arc is an arc-like [[Homogeneous space|homogeneous]] continuum. [[R.H. Bing]] proved that, in a certain well-defined sense, most continua in '''R'''<sup>''n''</sup>, ''n'' ≥ 2, are homeomorphic to the pseudo-arc.
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| == History ==
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| In 1920, [[Bronisław Knaster]] and [[Kazimierz Kuratowski]] asked whether a nondegenerate homogeneous continuum in the Euclidean plane '''R'''<sup>2</sup> must be a [[Jordan curve]]. In 1921, [[Stefan Mazurkiewicz]] asked whether a nondegenerate continuum in '''R'''<sup>2</sup> that is [[homeomorphic]] to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster descovered the first example of a homogeneous hereditarily indecomposable continuum ''K'', later named the pseudo-arc, giving a negative answer to the 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 E. Moise|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'''.<ref>{{harv|George W. Henderson|1960}} later showed that a ''decomposable'' continuum homeomorphic to all its nondenerate subcontinua must be an arc.</ref> 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]].<ref>The history of the discovery of the pseudo-arc is described in {{harv|Nadler|1992}}, pp 228–229.</ref>
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| == Construction ==
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| The following construction of the pseudo-arc follows {{harv|Wayne Lewis|1999}}.
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| === Chains ===
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| At the heart of the definition of the pseudo-arc is the concept of a ''chain'', which is defined as follows:
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| :A '''chain''' is a [[Finite set|finite collection]] of [[open set]]s <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 [[Element (mathematics)|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.
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| 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 [[Recursion|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.
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| More formally:
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| :Let <math>\mathcal{C}</math> and <math>\mathcal{D}</math> be chains such that
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| :# each link of <math>\mathcal{D}</math> is a subset of a link of <math>\mathcal{C}</math>, and
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| :# for any indices ''i'', ''j'', ''m'', and ''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> | |
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| :Then <math>\mathcal{D}</math> is '''crooked''' in <math>\mathcal{C}.</math>
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| ===Pseudo-arc===
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| For any collection ''C'' of sets, let <math>C^{*}</math> denote the union of all of the elements of ''C''. That is, let
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| :<math>C^*=\bigcup_{S\in C}S.</math>
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| The ''pseudo-arc'' is defined as follows:
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| :Let ''p'' and ''q'' be distinct points in the plane and <math>\left\{\mathcal{C}^{i}\right\}_{i\in\mathbb{N}}</math> be a sequence of chains in the plane such that for each ''i'',
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| :#the first link of <math>\mathcal{C}^i</math> contains ''p'' and the last link contains ''q'',
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| :#the chain <math>\mathcal{C}^i</math> is a <math>1/2^i</math>-chain,
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| :#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
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| :#the chain <math>\mathcal{C}^{i+1}</math> is crooked in <math>\mathcal{C}^i</math>.
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| :Let
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| ::<math>P=\bigcap_{i\in\mathbb{N}}\left(\mathcal{C}^i\right)^{*}.</math>
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| :Then ''P'' is a '''pseudo-arc'''.
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| == Notes ==
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| <references/>
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| ==References==
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| * [[R.H. Bing]], ''A homogeneous indecomposable plane continuum'', Duke Math. J., 15:3 (1948), 729–742
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| * R.H. Bing, ''Concerning hereditarily indecomposable continua'', Pacific J. Math., 1 (1951), 43–51
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| * George W. Henderson, ''Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc''. Annals of Math., 72 (1960), 421–428
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| * [[Bronisław Knaster]], ''Un continu dont tout sous-continu est indécomposable''. Fundamenta math. 3 (1922), 247–286
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| * Wayne Lewis, ''The Pseudo-Arc'', Bol. Soc. Mat. Mexicana, 5 (1999), 25–77
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| * [[Edwin Moise]], ''An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua'', Trans. Amer. Math. Soc., 63, no. 3 (1948), 581–594
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| * Sam B. Nadler, Jr, ''Continuum theory. An introduction''. Pure and Applied Mathematics, Marcel Dekker (1992) ISBN 0-8247-8659-9
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| [[Category:Continuum theory]]
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23 year-old Visual Arts and Crafts Professionals Toney Frogge from Atholville, loves people, ganhando dinheiro na internet and calligraphy. Discovered some lovely places after working 2 days at Angkor.
Here is my web site - como ficar rico