Closeness is a basic concept in topology and related areas in mathematics.
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.
is called close or near to a set
if where the distance between a point and a set is defined as where inf stands for infimum.
is called close to a set
is a closeness relation if it satisfies the following conditions: Let
[1] Topological spaces have a closeness relationship built into them: defining a point
Likewise, given a set with a closeness relation, defining a point
satisfies the Kuratowski closure axioms.
Thus, defining a closeness relation on a set is exactly equivalent to defining a topology on that set.
The closeness relation between a set and a point can be generalized to any topological space.
Given a topological space and a point
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.