The following construction of the pseudo-arc follows .
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
-chain if each of its links has
diameter less than . 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 -th link of the larger chain to the -th, the smaller chain must first move in a crooked manner from the -th link to the -th link, then in a crooked manner to the -th link, and then finally to the -th 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 with D_i\cap C_m\ne\emptyset, D_j\cap C_n\ne\emptyset, and m, there exist indices k and \ell with i (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 of sets, let denote the union of all of the elements of . That is, let :C^*=\bigcup_{S\in C}S. The
pseudo-arc is defined as follows: :Let be distinct points in the plane and \left\{\mathcal{C}^{i}\right\}_{i\in\N} be a sequence of chains in the plane such that for each , :#the first link of \mathcal{C}^i contains and the last link contains , :#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 is a
pseudo-arc. ==Notes==