Closed Sets

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Related to Closed Sets: Bounded set
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.


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


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
By (4) [??] = {[[bar.H].sub.1]: H [member of] F} separates closed sets. Hence there are sets [F.sub.1], [F.sub.2] [member of] F such that [f.sup.-1] ([[bar.G].sub.1]) [subset] ([[??].sub.1]), [f.sup.-1] ([[bar.G].sub.2]) [subset] ([[??].sub.2]) and [[??].sub.1] [intersection] [[??].sub.2] = [phi].
a Generalized Closed Sets in Neutrosophic Topological Spaces, International Journal of Mathematics Trends and Technology, Conference Series, (2018), 88-91.
Then X-U, X-V are disjoint closed sets. Since X is vg-normal there exist disjoint vg-open sets [U.sub.1] and [V.sub.1] such that X - U [subset] [U.sub.1] and X - V [subset] [V.sub.1].
(i) v is inner regular by closed sets and outer regular by open sets;
(1) g-continuous[3] if [f.sup.-1](V) is g-closed in (X, [tau]) for every closed set V of (Y, [sigma]).
Many real phenomena may be modelled as random closed sets in [R.sup.d], and in several situations as evolving random closed sets.
(iii) The union of any two neutrosophic soft cubic closed sets is a neutrosophic soft cubic closed set over X.
In this paper, we define and study the properties of [alpha] generalized regular weakly closed sets ([alpha]grw-closed) in topological space which is properly placed between the regular weakly closed sets and generalized pre regular weakly closed sets.
In this work, we have introduced some new notions of N-neutrosophic crisp open (closed) sets called []-semi (open) closed sets, []-preopen (closed) sets, and []-[alpha]-open (closed) sets and studied some of their basic properties in the context of neutrosophic crisp topological spaces.
Levine [12] introduced the concept of generalized closed sets in topological spaces and then Noiri and Popa [16] had studied it in detail.
Levine [5] introduced generalised closed sets and studied thier properties.
The complement of [F.sub.N]-open sets in the FNTS (X, [[tau].sub.n]) are called fuzzy neutrosophic closed sets ([F.sub.N]-closed set, for short).