Pseudo-arc

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.^[a] 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.^[b] 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 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

{\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{n}\}

in a metric space such that

C_{i}\cap C_{j}\neq \emptyset

if and only if

|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 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

{\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, and n with $D_{i}\cap C_{m}\neq \emptyset$ , $D_{j}\cap C_{n}\neq \emptyset$ , 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

{\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 and q be distinct points in the plane and

\left\{{\mathcal {C}}^{i}\right\}_{i\in \mathbb {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

^ Henderson (1960) later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.
^ The history of the discovery of the pseudo-arc is described in Nadler (1992), pp. 228–229.

References

Bing, R.H. (1948), "A homogeneous indecomposable plane continuum", Duke Mathematical Journal, 15 (3): 729–742, doi:10.1215/S0012-7094-48-01563-4
Bing, R.H. (1951), "Concerning hereditarily indecomposable continua", Pacific Journal of Mathematics, 1: 43–51, doi:10.2140/pjm.1951.1.43
Bing, R.H.; Jones, F. Burton (1959), "Another homogeneous plane continuum", Transactions of the American Mathematical Society, 90 (1): 171–192, doi:10.1090/S0002-9947-1959-0100823-3
Henderson, George W. (1960), "Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc", Annals of Mathematics, 2nd series, 72 (3): 421–428, doi:10.2307/1970224
Hoehn, Logan C.; Oversteegen, Lex G. (2016), "A complete classification of homogeneous plane continua", Acta Mathematica, 216 (2): 177–216, arXiv:1409.6324, doi:10.1007/s11511-016-0138-0
Hoehn, Logan C.; Oversteegen, Lex G. (2020), "A complete classification of hereditarily equivalent plane continua", Advances in Mathematics, 368: 107131, arXiv:1812.08846, doi:10.1016/j.aim.2020.107131
Irwin, Trevor; Solecki, Sławomir (2006), "Projective Fraïssé limits and the pseudo-arc", Transactions of the American Mathematical Society, 358 (7): 3077–3096, doi:10.1090/S0002-9947-06-03928-6
Kawamura, Kazuhiro (2005), "On a conjecture of Wood", Glasgow Mathematical Journal, 47 (1): 1–5, doi:10.1017/S0017089504002186
Knaster, Bronisław (1922), "Un continu dont tout sous-continu est indécomposable", Fundamenta Mathematicae, 3: 247–286, doi:10.4064/fm-3-1-247-286
Lewis, Wayne (1999), "The Pseudo-Arc", Boletín de la Sociedad Matemática Mexicana, 5 (1): 25–77
Lewis, Wayne; Minc, Piotr (2010), "Drawing the pseudo-arc" (PDF), Houston Journal of Mathematics, 36: 905–934
Moise, Edwin (1948), "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua", Transactions of the American Mathematical Society, 63 (3): 581–594, doi:10.1090/S0002-9947-1948-0025733-4
Nadler, Sam B. Jr. (1992), Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, ISBN 0-8247-8659-9
Rambla, Fernando (2006), "A counterexample to Wood's conjecture", Journal of Mathematical Analysis and Applications, 317 (2): 659–667, doi:10.1016/j.jmaa.2005.07.064
Rempe-Gillen, Lasse (2016), Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture, arXiv:1610.06278

[1] Henderson (1960) later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.

[2] The history of the discovery of the pseudo-arc is described in Nadler (1992), pp. 228–229.

[a]

[b]