Closeness (mathematics)

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In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topological spaces where we have no concrete way to measure distances.

The closure operator closes a given set by mapping it to a closed set which contains the original set and all points close to it. The concept of closeness is related to limit point.

Definition[edit]

Given a metric space (X,d) a point p is called close or near to a set A if

d(p,A) = 0,

where the distance between a point and a set is defined as

d(p, A) := \inf_{a \in A} d(p, a).

Similarly a set B is called close to a set A if

d(B,A) = 0

where

d(B, A) := \inf_{b \in B} d(b, A).

Properties[edit]

Closeness relation between a point and a set[edit]

Let A and B be two sets and p a point.

  • if p is close to A then A \neq \emptyset
  • if p is close to A and B \supset A then p is close to B
  • if p is close to A \cup B then either p is close to A or p is close to B

Closeness relation between two sets[edit]

Let A,B and C be sets.

  • if A and B are close then A \neq \emptyset and B \neq \emptyset
  • if A and B are close then B and A are close
  • if A and B are close and B \subset C then A and C are close
  • if A and B \cup C are close then either A and B are close or A and C are close
  • if A \cap B \neq \emptyset then A and B are close

Generalized definition[edit]

The closeness relation between a set and a point can be generalized to any topological space. Given a topological space and a point p, p is called close to a set A if p \in \operatorname{cl}(A) = \overline A.

To define a closeness relation between two sets the topological structure is too weak and we have to use a uniform structure. Given a uniform space, sets A and B are called close to each other if they intersect all entourages, that is, for any entourage U, (A×B)∩U is non-empty.

See also[edit]