Closed Sets


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Related to Closed Sets: Bounded set

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

References in periodicals archive ?
delta]]-[alpha]-locally closed sets and decreasing [[tau].
i) The intersection of any IF-semi closed sets is also IF-semi closed.
Noiri: A unified theory of generalized closed sets, The 6-th Meeting of Topological Spaces Theory and Their Applications, Aug.
Lahiri, Semi-generalized closed sets in topology, Indian J.
2009) the notion of local mean n-dimensional Minkowski content of a random closed set has been introduced in order to provide approximations of the mean density of ^-dimensional random closed sets in [R.
alpha]]:[alpha][member of] [DELTA]} be any pairwise regular cover of X by regularly closed sets in [[tau].
The subset A is semi closed if X\A is semi open and the semi closure of B, denoted by sclB, is the intersection of all semi closed sets containing B(Crossley, 1971).
ii) The inverse image of a neutrosophic closed sets in Y is a neutrosophic semi closed set in X.
Norman Levine introduced generalized closed sets, K.
We define as the union of all open subsets of and as the intersection of all closed sets containing .
The elements of the [sigma]-algebra generated by zero-sets are called Baire sets and the elements of the [sigma]-algebra generated by closed sets are called Borel sets; B(X)and [B.
In [4] Frink's procedure uses a normal base of closed sets instead of the family of all closed sets as employed by Wallman (6).