open set

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open set

n
1. (Logic) a set which is not a closed set
2. (Mathematics) an interval on the real line excluding its end points, as [0, 1], the set of reals between, but excluding, 0 and 1
References in classic literature ?
Those who remarked in the physiognomy of the Prince a dissolute audacity, mingled with extreme haughtiness and indifference to, the feelings of others could not yet deny to his countenance that sort of comeliness which belongs to an open set of features, well formed by nature, modelled by art to the usual rules of courtesy, yet so far frank and honest, that they seemed as if they disclaimed to conceal the natural workings of the soul.
After that, Khedr [4] introduced and studied about [[tau].sub.1][[tau].sub.2]- open sets. Later, Fukutake [5] defined one kind of semiopen sets in 1989.
Levine [5] defined semi open sets in topological spaces.
Now B = ((X - G) [union] B) [intersection] G is the intersection of two open sets, so that B is open.
Let (X, [tau], E) be a soft topological space over X; then the members of [tau] are said to be soft open sets in X.
Another approach is the study of the topologies by their number of open sets. For this, let T(n, k) be the number of all labeled topologies with k open sets that we can define on E and let [T.sub.0](n, k) be the number of those which are [T.sub.0].
Weak forms of soft open sets were first studied by Chen [5].
By the fact that Y is [T.sub.2], there exist open sets W and V such that f(x) [member of] W, y [member of] V and V [intersection] W = [phi].
Csaszar [1] has introduced and studied generalized open sets of a set X denned in terms of monotonic functions [gamma] : P(X) [right arrow] P(X).
Csaszar: Generalized open sets in generalized topologies, Acta Math.
Then C = {x: there exist semiopen sets A and B such that {x}= A [intersection] B}= D = {x:{x}[member of] T}[union]{x: there exist disjoint open sets U and V such that x[member of] Cl(U) [intersection] Cl(V)}.
Hence, all open sets in Y are of the form g(V)\g(X\V), V is sg-open in X.

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