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 ) {\displaystyle (X,d)} is said to be well-chained if for every x , y ∈ X {\displaystyle x,y\in X} and every ε > 0 {\displaystyle \varepsilon >0} there exists n ∈ N {\displaystyle n\in \mathbb {N} } and z 0 , z 1 , … , z n ∈ X {\displaystyle z_{0},z_{1},\dotsc ,z_{n}\in X} such that z 0 = x {\displaystyle z_{0}=x} , z n = y {\displaystyle z_{n}=y} and for every j ∈ { 1 , … , n − 1 } {\displaystyle j\in \{1,\dotsc ,n-1\}} , one has
d ( z j − 1 , z j ) < ε {\displaystyle d(z_{j-1},z_{j})<\varepsilon } . A set A ⊆ X {\displaystyle A\subseteq X} is well-chained if it is well-chained as a metric space with the distance d {\displaystyle d} restricted to A {\displaystyle A} .
Properties A set A ⊆ X {\displaystyle A\subseteq X} is well-chained if and only if its topological closure is well-chained. If X {\displaystyle X} is well-chained and if f : X → Y {\displaystyle f\colon X\to Y} is uniformly continuous then the set f ( X ) {\displaystyle f(X)} is well-chained.
Characterizations The following properties are equivalent:
the space X {\displaystyle X} is well-chained; if A ⊆ X {\displaystyle A\subseteq X} and ∅ ≠ A ≠ X {\displaystyle \emptyset \neq A\neq X} , then inf { d ( x , y ) : x ∈ A and y ∈ X ∖ A } = 0 {\displaystyle \inf\{d(x,y):x\in A{\text{ and }}y\in X\setminus A\}=0} ; if f : X → { 0 , 1 } {\displaystyle f\colon X\to \{0,1\}} is uniformly continuous, then f {\displaystyle f} is constant.
Link with connectedness Any well-chained set X {\displaystyle X} is connected. The converse fails in general:
the set of rational numbers Q {\displaystyle \mathbb {Q} } is well-chained but not connected, the set { ( x , y ) ∈ R 2 : x 2 y 2 = x y } {\displaystyle \{(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 ⊆ R {\displaystyle A\subseteq \mathbb {R} } is closed and well-chained, then A {\displaystyle 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.
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