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

1. The complement of is an open set ,

2. is its own set
closure ,

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

4. Every point outside
has a neighborhood disjoint from .

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

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 .

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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References 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. Referenced
on Wolfram|Alpha Closed Set
Cite this as:
Weisstein, Eric W. "Closed Set." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ClosedSet.html

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