Closure of a set

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in a topological space

The intersection of all closed sets (cf. Closed set) containing the set.


In the Russian literature the closure of a set is denoted by , or to express that the closure is taken in the space , in the Western literature one uses , , , or .

Another definition of closure is as follows. The closure of in is the set of all satisfying: Every neighbourhood of intersects .

The closure operation satisfies: 1) ; 2) ; 3) ; and 4) . Any operation satisfying 1), 2), 3), and 4) is called a closure operation. One can define a topological space by means of a closure operation: The closed sets are to be those sets that equal their own closure.

This approach is taken in [a1].


[a1] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
How to Cite This Entry:
Closure of a set. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article