---
tag: VGCTDCGS6V
kind: example
name: Pseudo-arc
msc: [54F15, 54F50]
instance_of: [56J7DSVAZH]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo-arc
same_as:
  wikidata: Q4382843
  wikipedia: Pseudo-arc
---

## Example

A [[continuum|NJQ196M1WR]] $X$ 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|56J7DSVAZH]].
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.

## Remarks

The pseudo-arc is homogeneous: for any two points there is a homeomorphism of the space onto itself that sends one to the other.
It is homeomorphic to each of its nondegenerate subcontinua, as Moise proved.
The arc $[0, 1]$ is also chainable and also homeomorphic to its nondegenerate subcontinua, but it is decomposable; this is why the name "pseudo-arc" fits.

## Explanations

### L1 human

The pseudo-arc looks like an arc from far away, since it can be covered by a thin chain of small links.
Up close it is folded back on itself at every scale, so that no piece of it is a simple arc.
Every piece of it is as tangled as the whole, and every piece looks the same as the whole.