Pseudo-arc

In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. 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.

Contents

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 described the first example of a homogeneous hereditarily indecomposable continuum K. In 1948, R.H. Bing constructed a continuum B giving a negative answer to the Knaster–Kuratowski question and Edwin Moise constructed a continuum M giving a negative answer to the Mazurkiewicz question. 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 and showed that it was hereditarily indecomposable.[1] Bing's construction of B 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.[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\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 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
  1. each link of \mathcal{D} is a subset of a link of \mathcal{C}, and
  2. for any indices i, j, m, and n with D_i\cap C_m\ne\emptyset, D_j\cap C_n\ne\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%2B1}.
Then \mathcal{D} is crooked in \mathcal{C}.

Pseudo-arc

For any collect 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,
  1. the first link of \mathcal{C}^i contains p and the last link contains q,
  2. the chain \mathcal{C}^i is a 1/2^i-chain,
  3. the closure of each link of \mathcal{C}^{i%2B1} is a subset of some link of \mathcal{C}^i, and
  4. the chain \mathcal{C}^{i%2B1} 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

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

References