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.

A metric space ${\displaystyle (X,d)}$ is said to be well-chained if for every ${\displaystyle x,y\in X}$ and every ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle z_{0},z_{1},\dotsc ,z_{n}\in X}$ such that ${\displaystyle z_{0}=x}$, ${\displaystyle z_{n}=y}$ and for every ${\displaystyle j\in \{1,\dotsc ,n-1\}}$, one has ${\displaystyle d(z_{j-1},z_{j})<\varepsilon }$.^{1: Ch. I §8 2}.

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

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

If ${\displaystyle X}$ is well-chained and if ${\displaystyle f\colon X\to Y}$ is uniformly continuous then the set ${\displaystyle f(X)}$ is well-chained².

The following properties are equivalent²:

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

Any well-chained set ${\displaystyle X}$ is connected ^{1: Ch. I §8}.

The converse fails in general:

• the set of rational numbers ${\displaystyle \mathbb {Q} }$ is well-chained but not connected ^1: §I.8,
• the set ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x^{2}y^{2}=xy\}}$ 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 ${\displaystyle A\subseteq \mathbb {R} }$ is closed and well-chained, then ${\displaystyle A}$ is connected².

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.