Pseudo-arc

In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum. 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 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 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.^{[lower-alpha 1]} 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.^{[lower-alpha 2]}

Construction

The following construction of the pseudo-arc follows (Wayne 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.

References

Notes

↑ George W. Henderson later showed that a decomposable continuum homeomorphic to all its nondegenerate subcontinua must be an arc.^[1]
↑ The history of the discovery of the pseudo-arc is described in ,^[2] pp 228–229.

Citations

Bibliography

R.H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J., 15:3 (1948), 729–742
R.H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math., 1 (1951), 43–51
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 Math. 72: 421–428.
Bronisław Knaster, Un continu dont tout sous-continu est indécomposable. Fundamenta Mathematicae 3 (1922): pp. 247–286
Wayne Lewis, The Pseudo-Arc, Bol. Soc. Mat. Mexicana, 5 (1999), 25–77
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
Sam B. Nadler, Jr, Continuum theory. An introduction. Pure and Applied Mathematics, Marcel Dekker (1992) ISBN 0-8247-8659-9

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