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 (X, d) is said to be well-chained if for every x, y \in X and every \varepsilon > 0 there exists n \in \mathbb{N} and z_0, z_1, \dotsc, z_n \in X such that z_0= x, z_n = y and for every j \in \{1, \dotsc, n - 1\}, one has d (z_{j - 1}, z_j) < \varepsilon.[1]: Ch. I §8[2].

A set A \subseteq X is well-chained if it is well-chained as a metric space with the distance d restricted to A.

Properties

A set A \subseteq X is well-chained if and only if its topological closure is well-chained.

If X is well-chained and if f \colon X \to Y is uniformly continuous then the set f (X) is well-chained[2].

Characterizations

The following properties are equivalent[2]:

  1. the space X is well-chained;
  2. if A \subseteq X and \emptyset \ne A \ne X, then \inf \{ d (x, y) : x \in A \text{ and } y \in X \setminus A\} = 0;
  3. if f \colon X \to \{0, 1\} is uniformly continuous, then f is constant.

Any well-chained set X is connected [1]: Ch. I §8.

The converse fails in general:

  • the set of rational numbers \mathbb{Q} is well-chained but not connected [1]: §I.8,
  • the set \{(x, y) \in \mathbb{R}^2 : x^2 y^2 = xy\} is well-chained but not connected[3]: § 33.

There are some situations where well-chainedness implies connectedness:

  • every compact and well-chained set is connected [1]: (I.9.21);
  • if A \subseteq \mathbb{R} is closed and well-chained, then A is connected[2].

History

The definition of well-chained space was proposed as a definition of connected space (zusammenltiengende Punktmenge) by Georg Cantor in 1883[4]: §11.

In 1921, Maurice Fréchet names well-chained set (ensemble bien enchaîné) connected sets and proves, in the current terminology, that connected spaces are well-chained spaces[3]: §33.

The definition above appears in 1964 under the name of well-chained space in the book of Gordon Whyburn [1]: §I.8.

References

  1. ^ Whyburn, Gordon Thomas (1964). Topological Analysis. 2 ed. Princeton, N.J.: Princeton University Press.
  2. ^ Mathews, Jerold C. (March 1968). "A note on well-chained spaces". The American Mathematical Monthly. 75 (3): 273. doi:10.2307/2314959
  3. ^ Fréchet, Maurice (1921). "Sur les ensembles abstraits". Annales scientifiques de l'École normale supérieure. 38: 341–388. doi:10.24033/asens.735
  4. ^ Cantor, Georg (December 1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/BF01446819