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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.
Given a metric space a point is called close or near to a set if
where the distance between a point and a set is defined as
Similarly a set is called close to a set if
- if a point is close to a set and a set then and are close (the converse is not true!).
- closeness between a point and a set is preserved by continuous functions
- closeness between two sets is preserved by uniformly continuous functions
Closeness relation between a point and a set
Let and be two sets and a point.
- if is close to then
- if is close to and then is close to
- if is close to then either is close to or is close to
Closeness relation between two sets
Let , and be sets.
- if and are close then and
- if and are close then and are close
- if and are close and then and are close
- if and are close then either and are close or and are close
- if then and are close
The closeness relation between a set and a point can be generalized to any topological space. Given a topological space and a point , is called close to a set if .
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.