Closed Sets

Closed Sets 

in mathematics, point sets on a straight line, on a plane, or in space, containing all their points of tangency. The point of tangency of the set £ is a point (which need not belong to E) such that in any neighborhood of the point there is at least one point of E. A geometric figure (a circle, square), with its boundary points included, can serve as an example of a closed set. The union of a finite number and the intersection of any number of a closed set will again be a closed set. The complement of any closed set is an open set and vice versa. Along with open sets, closed sets are the simplest types of point sets and play an important role in the theory of functions and, in particular, the theory of measure. Among closed sets, particularly notable, owing to their re-markable qualities, are perfect sets, that is, closed sets that do not have isolated points.

The definition of a closed set also holds for sets in arbitrary metric and topological spaces. For sets in metric spaces it is equivalent to the fact that a closed set is a set containing all its limit points.

REFERENCES

Aleksandrov, P. S. Vvedenie v obshchuiu teoriiu mnozhestv i funktsii. Moscow-Leningrad, 1948. Rudin, W. Osnovy matematicheskogo analiza. Moscow, 1966. (Translated from English.)

S. B. STECHKIN