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.

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.

References in periodicals archive ?
26]), which asserts that every nonempty relatively weakly open subset of [B.
Let U be a nonempty relatively weakly open subset of [B.
Let Z be a Stein complex manifold of dimension n, and Y a Runge open subset of Z.
i]) is pairwise almost regular Lindelof iff every pairwise (quasi) regularly open subset of X having the countable intersection property has a non-empty intersection.
1] be the family of all (1,2)-regular open subsets of X and let [B.
If D is a non-empty simply connected open subset of the complex plane C which is not all of C, then there exists a biholomorphic (bijective and holomorphic) mapping f from D onto the open unit disk U = {z [member of] C :[absolute value of z] <1} (Krantz, 1999, Section 6.
Conversely, let B be a regular open subset in (Y, [K.
Assume that x * y [member of] G, with x, y [member of] A and G an open subset of A.
U [union] V : U is an open subset of Y and V is an open subset of Z} is the topology of X.
It is easily shown that the semi-open subsets of Fare the open subsets of Y along with {0,1} and {0,2}.
lambda]]: [lambda] [member of] [LAMBDA]} of open subsets of X such that A [subset] [union] {[G.
Then there exist disjoint non-empty regular open subsets [B.

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