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.
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This article is about the relation between two sets. For the description a single set, see
Closedness.
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.
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
where inf stands for infimum. Similarly a set is called close to a set if
where
- .
- 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
Arkhangel'skii, A. V.; Pontryagin, L.S. General Topology I: Basic Concepts and Constructions, Dimension Theory. Encyclopaedia of Mathematical Sciences (Book 17), Springer 1990, p. 9.