Pseudo-arc

Example

A continuum $X$ is chainable if for every $\varepsilon > 0$ there is a finite cover of $X$ by open sets $U_1, \dots, U_n$ of the following kind. Each $U_i$ has diameter less than $\varepsilon$, and $U_i$ meets $U_j$ only when $\abs{i - j} \le 1$. The pseudo-arc is a nondegenerate chainable continuum in which every subcontinuum is indecomposable. It can be built in the plane as the intersection of a decreasing sequence of chains of discs. Each chain is folded back and forth inside the previous one in a specific crooked way. Knaster gave the first such continuum, and Bing proved that any two such continua are homeomorphic. The pseudo-arc is indecomposable, since by its definition every subcontinuum, including the whole space, is indecomposable.