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 $\varepsilon >0$ 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]

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

Link with connectedness

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 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.

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