Well-chained space


In mathematics, a well-chained space is a metric space in which two arbitrary points can be connected by a chain of points that are arbitrarily close. It is closely related to the notion of connectedness.

Formal definition

A metric space is said to be well-chained if for every and every there exists and such that, and for every, one has
..
A set is well-chained if it is well-chained as a metric space with the distance restricted to.

Properties

A set is well-chained if and only if its topological closure is well-chained.
If is well-chained and if is uniformly continuous then the set is well-chained.

Characterizations

The following properties are equivalent:
  1. the space is well-chained;
  2. if and, then ;
  3. if is uniformly continuous, then is constant.

Link with connectedness

Any well-chained set is connected.
The converse fails in general:
There are some situations where well-chainedness implies connectedness:
  • every compact and well-chained set is connected ;
  • if is closed and well-chained, then is connected.

History

The definition of well-chained space was proposed as a definition of connected space by Georg Cantor in 1883.
In 1921, Maurice Fréchet names well-chained set connected sets and proves, in the current terminology, that connected spaces are well-chained spaces.
The definition above appears in 1964 under the name of well-chained space in the book of Gordon Whyburn.