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: the space is well-chained; if and , then ; if is uniformly continuous, then is constant.
Link with connectedness
Well-chainedness is a generalization of connectedness, in the sense that any connected (metric) space is well-chained.
The converse fails in general: the set of rational numbers is well-chained but not connected, the set is well-chained but not connected.
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.
References