Closed Set


There are several equivalent definitions of a closed set. Let S be a subset of a metric space. A set S is closed if

1. The complement of S is an open set,

2. S is its own set closure,

3. Sequences/nets/filters in S that converge do so within S,

4. Every point outside S has a neighborhood disjoint from S.

The point-set topological definition of a closed set is a set which contains all of its limit points. Therefore, a closed set C is one for which, whatever point x is picked outside of C, x can always be isolated in some open set which doesn't touch C.

The most commonly encountered closed sets are the closed interval, closed path, closed disk, interior of a closed path together with the path itself, and closed ball. The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points (and is nowhere dense, so it has Lebesgue measure 0).

It is possible for a set to be neither open nor closed, e.g., the half-closed interval (0,1].

See also

Borel Set, Boundary Point, Cantor Set, Closed Ball, Closed Interval, Closed Disk, Compact Set, Half-Closed Interval, Open Set Explore this topic in the MathWorld classroom

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Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 3, 1999.

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Closed Set

Cite this as:

Weisstein, Eric W. "Closed Set." From MathWorld--A Wolfram Web Resource.

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