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 $(X, d)$ is said to be well-chained if for every $x, y \in X$ and every $ε > 0$ there exists $n \in ℕ$ and $z_{0}, z_{1}, \dots, z_{n} \in X$ such that $z_{0} = x$ , $z_{n} = y$ and for every $j \in {1, \dots, n - 1}$ , one has $d (z_{j - 1}, z_{j}) < ε$ .^{: Ch. I §8}

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 $f : X \to Y$ is uniformly continuous then the set $f (X)$ is well-chained.

Characterizations

The following properties are equivalent:

the space $X$ is well-chained;
if $A \subseteq X$ and $∅︀ \neq A \neq X$ , then $inf {d (x, y) : x \in A and y \in X ∖ A} = 0$ ;
if $f : X \to {0, 1}$ is uniformly continuous, then $f$ is constant.

Link with connectedness

Well-chainedness is a generalization of connectedness, in the sense that any connected (metric) space $X$ is well-chained.^{: Ch. I §8}

The converse fails in general:

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

There are some situations where well-chainedness implies connectedness:

every compact and well-chained set is connected;^: (I.9.21)
if $A \subseteq ℝ$ is closed and well-chained, then $A$ is connected.

History

The definition of well-chained space was proposed as a definition of connected space (zusammenltiengende Punktmenge) by Georg Cantor in 1883.^: §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.^: §33

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

References

1 2 3 4 5 Whyburn, Gordon Thomas (1964). Topological Analysis (2 ed.). Princeton, N.J.: Princeton University Press.
1 2 3 4 Mathews, Jerold C. (March 1968). "A note on well-chained spaces". The American Mathematical Monthly. 75 (3): 273. doi:10.2307/2314959.
1 2 Fréchet, Maurice (1921). "Sur les ensembles abstraits". Annales scientifiques de l'École normale supérieure. 38: 341–388. doi:10.24033/asens.735.
↑ Cantor, Georg (December 1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/BF01446819.

[Whyburn_1964-1] 1 2 3 4 5 Whyburn, Gordon Thomas (1964). Topological Analysis (2 ed.). Princeton, N.J.: Princeton University Press.

[Mathews_1968-2] 1 2 3 4 Mathews, Jerold C. (March 1968). "A note on well-chained spaces". The American Mathematical Monthly. 75 (3): 273. doi:10.2307/2314959.

[Frechet_1921-3] 1 2 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.