Open Set

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open set

[′ō·pən ‚set]
(mathematics)
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 It might be outdated or ideologically biased.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 ?
(3) Collection of all [[tau].sub.1]-[delta] open sets and [[tau].sub.2]-[delta] open sets are denoted by [[tau].sub.1s] and [[tau].sub.2s] respectively.
i.e if there do not exist disjoint semi open sets (Eq.) and (Eq.) of (Eq.) , such that (Eq.).
Clearly cl((F, A)) is the smallest soft closed set over X which contains (F, A) and int((F, A)) is the largest soft open set over X which is contained in (F, A).
For the unlabelled case, let t(n, k) be the number of all unlabeled topologies having k open sets, let [t.sub.0](n, k) be the number of those which are [T.sub.0], and let [t.sub.n0](n, k) be the number of those which are non-[T.sub.0].
Clearly cl(A, E) is the smallest soft closed set over X which contains (A, E) and int(A, E) is the largest soft open set over X which is contained in (A, E).
The generalized closure of a subset S of X, denoted by [c.sub.[mu]](S), is the intersection of generalized closed sets including S and the interior of S, denoted by [i.sub.[mu]](S), is the union of generalized open sets contained in S.
Min: Generalized continuous functions defined by generalized open sets on generalized topological spaces, Acta Math.
Let U and V be open sets such that U [subset or equal to]A [subset or equal to] Cl(U) and V [subset or equal to] B [subset or equal to]Cl(V).
A topological space (X, [tau]) is said to be sg*-normal if for any pair of disjoint sg-closed subsets F1 and F2 of X, there exist disjoint open sets U and V such that F1 [subset] U and F2 [subset] V .
Njastad, On some classes of nearly open sets, Pacific J.
Recall that a Hausdorff space is said to be regular if for each closed subset F of X, p [member of] X\F, there exist disjoint open sets U, V such that p [member of] U, F [subset] V.

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