Pseudo-arc

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 ${\displaystyle {\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{n}\}}$ in a metric space such that ${\displaystyle C_{i}\cap C_{j}\neq \emptyset }$ if and only if ${\displaystyle |i-j|\leq 1.}$ 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 ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$ be chains such that

1. each link of ${\displaystyle {\mathcal {D}}}$ is a subset of a link of ${\displaystyle {\mathcal {C}}}$, and
2. for any indices i, j, m, and n with ${\displaystyle D_{i}\cap C_{m}\neq \emptyset }$, ${\displaystyle D_{j}\cap C_{n}\neq \emptyset }$, and ${\displaystyle m<n-2}$, there exist indices ${\displaystyle k}$ and ${\displaystyle \ell }$ with ${\displaystyle i<k<\ell <j}$ (or ${\displaystyle i>k>\ell >j}$) and ${\displaystyle D_{k}\subseteq C_{n-1}}$ and ${\displaystyle D_{\ell }\subseteq C_{m+1}.}$

Then ${\displaystyle {\mathcal {D}}}$ is crooked in ${\displaystyle {\mathcal {C}}.}$

For any collection C of sets, let ${\displaystyle C^{*}}$ denote the union of all of the elements of C. That is, let

${\displaystyle C^{*}=\bigcup _{S\in C}S.}$

The pseudo-arc is defined as follows:

Let p and q be distinct points in the plane and ${\displaystyle \left\{{\mathcal {C}}^{i}\right\}_{i\in \mathbb {N} }}$ be a sequence of chains in the plane such that for each i,

1. the first link of ${\displaystyle {\mathcal {C}}^{i}}$ contains p and the last link contains q,
2. the chain ${\displaystyle {\mathcal {C}}^{i}}$ is a ${\displaystyle 1/2^{i}}$-chain,
3. the closure of each link of ${\displaystyle {\mathcal {C}}^{i+1}}$ is a subset of some link of ${\displaystyle {\mathcal {C}}^{i}}$, and
4. the chain ${\displaystyle {\mathcal {C}}^{i+1}}$ is crooked in ${\displaystyle {\mathcal {C}}^{i}}$.

Let

${\displaystyle P=\bigcap _{i\in \mathbb {N} }\left({\mathcal {C}}^{i}\right)^{*}.}$

Then P is a pseudo-arc.