Well-chained space

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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)} {\displaystyle (X,d)} is said to be well-chained if for every x , y ∈ X {\displaystyle x,y\in X} {\displaystyle x,y\in X} and every ε > 0 {\displaystyle \varepsilon >0} {\displaystyle \varepsilon >0} there exists n ∈ N {\displaystyle n\in \mathbb {N} } {\displaystyle n\in \mathbb {N} } and z 0 , z 1 , … , z n ∈ X {\displaystyle z_{0},z_{1},\dotsc ,z_{n}\in X} {\displaystyle z_{0},z_{1},\dotsc ,z_{n}\in X} such that z 0 = x {\displaystyle z_{0}=x} {\displaystyle z_{0}=x}, z n = y {\displaystyle z_{n}=y} {\displaystyle z_{n}=y} and for every j ∈ { 1 , … , n − 1 } {\displaystyle j\in \{1,\dotsc ,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 } {\displaystyle d(z_{j-1},z_{j})<\varepsilon }.[1]: Ch. I §8 [2].

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

Properties

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

If X {\displaystyle X} {\displaystyle X} is well-chained and if f : X → Y {\displaystyle f\colon X\to Y} {\displaystyle f\colon X\to Y} is uniformly continuous then the set f ( X ) {\displaystyle f(X)} {\displaystyle f(X)} is well-chained[2].

Characterizations

The following properties are equivalent[2]:

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

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

The converse fails in general:

  • the set of rational numbers Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} } is well-chained but not connected [1]: §I.8 ,
  • the set { ( x , y ) ∈ R 2 : x 2 y 2 = x y } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x^{2}y^{2}=xy\}} {\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 A ⊆ R {\displaystyle A\subseteq \mathbb {R} } {\displaystyle A\subseteq \mathbb {R} } is closed and well-chained, then A {\displaystyle A} {\displaystyle 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. Whyburn, Gordon Thomas (1964). Topological Analysis (2 ed.). Princeton, N.J.: Princeton University Press.
  2. Mathews, Jerold C. (March 1968). "A note on well-chained spaces". The American Mathematical Monthly. 75 (3): 273. doi:10.2307/2314959.
  3. 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.