Open Set

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Related to Open Set: Closed set, Connected set

open set

[′ō·pən ‚set]
A set included in a topology; equivalently, a set which is a neighborhood of each of its points; a topology on a space is determined by a collection of subsets which are called open.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Open Set


a point set that does not contain the limit points of its complement. Every point of an open set is an interior point, that is, it has a neighborhood entirely contained in the open set. Together with closed sets, open sets play an important role in the theory of functions, in topology, and in other branches of mathematics. Any nonempty open set on a line is an open interval or a sum of an at most countable number of open intervals.

The concept of open set can be applied in an n-dimensional Euclidean space and also in an arbitrary metric or topological space. The intersection of a finite number of open sets is an open set, as is the union of any number of open sets. Connected open sets are called domains. Any topological space can be defined by specifying its open sets. If a topological space is given by a system of its closed sets, then the open sets are defined in it as the complements of the closed sets.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
The complement of a g[ALEPH] closed set is called a g[ALEPH] open set.
(2) A is said to be [[tau].sub.2]-[delta] open set, if for x [member of] A, there exists [[tau].sub.21]-regular open set G such that x [member of] G [subset] A.
The complement of a semi open set is said to be semi closed, the semi closure of (Eq.) , denoted by (Eq.) is the intersection of all semi closed subsets of containing [12], [13].
Then, let us take (F, E) = {([e.sub.1], {[x.sub.1], [x.sub.3]}), ([e.sub.2], {[x.sub.2]}), ([e.sub.3], {[x.sub.1], [x.sub.2]})}; then int(F, E) = [??], int(cl(int((F, E)))) = [??], and so (F, E)c int(cl(int((F, E)))); hence, (F, E) is soft [alpha]-open set but not soft open set (since (F, E) is not soft open set).
The nonempty open set A is called a minimal open set if, for all O in [tau], we have
(2) for each soft singleton (P, E) in X and each soft open set (O,K) in Y and f((P, E))[??](O, K), there exists a soft [alpha]-open set (U, E) in X such that (P, E)[??](U, E) and f((U, E))[??](O, K);
Then the following are equivalent: (a) f:(X,T) [right arrow] (Y,S) is semi-strongly continuous, (b) f:(X,T) [right arrow] (Y,TSO(Y,S)) is open, closed, and continuous, (c) D= {[f.sup.-1] f(p)): p [member of] X} is a decomposition of (X,T) into closed, open sets, and (d) for each p [member of] X, there exists a closed, open set U containing p on which f is constant.
As Vy is a neighbourhood of y, there exists an open set Wy in Y such that y [member of] Wy [subset] Vy .
Then every open set of X is regular open if and only if every open set is closed.
Since that each interval of [0,[omega]] is an open set, we denote by [absolute value of n,m] one interval with endpoints n,m.
In 1965 Njastad [1] introduced a open sets.A subset of a topological space is called [alpha] open if A is a subset of int cl int(A).The complement of a open set is a closed.

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