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 ?
(2) f (resp., [phi]) is cofinitely sensitive in X if there is a constant [delta] > 0 such that [N.sub.f](V, [delta]) [contains] [n, +[infinity]) [intersection] N for some n [greater than or equal to] 1 (resp., [N.sub.[phi]] (V, [delta]) [contains] [t, +[infinity]) for some t > 0) for any nonempty open subset V [subset] X;
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 6.4.3, p.
It is now clear that g(D) can be considered as an open subset in H, here g and D are in the statement of our main theorem in Section 1.
Let X be a completely regular (respectively normal) topological space, V an open subset of X and let F [member of] [MA.sub.[partial derivative]V]([bar.V], X) be [PSI]-essential in [MA.sub.[partial derivative]V]([bar.V], X).
Let [OMEGA] be an open subset of [R.sup.N]; let L : [R.sup.N] [right arrow] [R.sup.+] be convex, and let the growth condition (GC) be satisfied.
Definition 1 A subset A of a space (X, T) is called [zeta]-open if for every x [member of] A , there exists an open subset U [subset not equal to] X containing x and such that U \ sInt(A) is countable.
Let E be a Hausdorff topological space and U an open subset of E.
Let M be ([M.sub.sc]) with [LAMBDA] = [R.sub.>0], let U [subset or equal to] E be a [c.sup.[infinity]]- open subset in a convenient vector space E and F be a Banach space.
[1, 5]) An admissible map [phi]: X [??] X is called a compact absorbing contraction (CAC-map for short) provided there exists an open subset U [subset] X such that:
An intuitionistic fuzzy set A of an IFTS(X, [tau]) is said to be intuitionistic fuzzy weakly generalized compact relative to X if every collection {[A.sub.i] : i [member of] I} of intuitionistic fuzzy weakly generalized open subset of X such that A [subset or equal to] [union] {[A.sub.i] : i [member of] I}, there exists a finite subset [I.sub.0] of I such that A [subset or equal to] [union] {[A.sub.i] : i [member of] [I.sub.0]} .
There exists an open subset [Y.sub.0] of Y := Spec [S.sup.G] such that [Mathematical Expression Omitted] is the preimage of [Y.sub.0] via the map V [approximate] Spec S [approaches] Spec [S.sup.G] = Y and the fibres of [Mathematical Expression Omitted] are precisely the G-orbits of G in [Mathematical Expression Omitted].
Let E be a completely regular (respectively normal) topological space, U an open subset of E, F [member of] [A.sub.[partial derivative]U] ([bar.U], E) and let G [member of] [A.sub.[partial derivative]U]([bar.U], E) be [phi]- essential in [A.sub.[partial derivative]U] ([bar.U], E).