Closed Set

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 .

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.