Closed Set
المؤلف:
Croft, H. T.; Falconer, K. J.; and Guy, R. K
المصدر:
Unsolved Problems in Geometry. New York: Springer-Verlag
الجزء والصفحة:
...
15-7-2021
2315
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 ![(0,1]](https://mathworld.wolfram.com/images/equations/ClosedSet/Inline14.gif)
.
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.
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