Well-chained space

Short description: Metric space connected by chains

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 $\epsilon >0$ there exists $n\in \mathbb{\mathbb{N}}$ 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 $d(z_{j-1},z_j)<\epsilon$.

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:

1. the space $X$ is well-chained;
2. if $A\subseteq X$ and $\mathrm{\varnothing }\ne A\ne X$, then $\inf \{d(x,y):x\in A\text{ and }y\in X\setminus A\}=0$;
3. 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.

The converse fails in general:

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

There are some situations where well-chainedness implies connectedness:

• every compact and well-chained set is connected;
• if $A\subseteq \mathbb{ℝ}$ 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.

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.

The definition above appears in 1964 under the name of well-chained space in the book of Gordon Whyburn.