Pseudo-arc

In point-set topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. It was first discovered, in 1922, by the renowned Polish topologist Bronislaw Knaster. The following definitions are, with slight modifications, due to Wayne Lewis (see the references section below). Other definitions have appeared in papers by R.H. Bing and Edwin Moise; they yield homeomorphic spaces.

Definitions

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 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 <math>\mathcal{C}</math> and <math>\mathcal{D}</math> be chains such that

1. each link of <math>\mathcal{D}</math> is a subset of a link of <math>\mathcal{C}</math>, and
2. 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 k and l with <math>i<k<l<j</math> (or <math>i>k>l>j</math>) and <math>D_k\subseteq C_{n-1}</math> and <math>D_l\subseteq C_{m+1}.</math>

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

Pseudo-arc

For any collect C of sets, let <math>C^{*}</math> denote the union of all of the elements of C. That is, let

<math>C^*=\bigcup_{S\in C}S.</math>

The pseudo-arc is defined as follows:

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,

1. the first link of <math>\mathcal{C}^i</math> contains p and the last link contains q,
2. the chain <math>\mathcal{C}^i</math> is a <math>1/2^i</math>-chain,
3. 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
4. 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 P is a pseudo-arc.

External links

• Pseudoarc for the People - Ongoing art project based on the pseudoarc

References

• Bing, R.H. 1948. A Homogeneous Indecomposable Plane Continuum, Duke Mathematical Journal Volume 15, no. 3, 729–742.
• Lewis, Wayne. 1999. The Pseudo-Arc, Bol. Soc. Mat. Mexicana Volume 5, 25–77.
• Moise, Edwin. 1948. An Indecomposable Plane Continuum Which is Homeomorphic to Each of Its Nondegenerate Subcontinua, Transactions of the American Mathematical Society Volume 63, no. 3, 581–594.