Notes on Introductory Point
... contained in O . The points f (x) that are not in O are therefore not in (c, d) so they remain at least a fixed positive distance from f (x0 ) . To summarize: there are points x arbitrarily close to x0 for which f (x) remains a fixed positive distance away from f (x0 ) . This certainly says that f i ...
... contained in O . The points f (x) that are not in O are therefore not in (c, d) so they remain at least a fixed positive distance from f (x0 ) . To summarize: there are points x arbitrarily close to x0 for which f (x) remains a fixed positive distance away from f (x0 ) . This certainly says that f i ...
On slightly I-continuous Multifunctions 1 Introduction
... Proof. Let x ∈ X, K ⊂ Y and H ⊂ Z be clopen sets such that x ∈ F1+ (K) and x ∈ F2+ (H). Then we obtain that F1 (x) ⊂ K and F2 (x) ⊂ H and thus, F1 (x) × F2 (x) = (F1 ×F2 )(x) ⊂ K ×H. We have x ∈ (F1 ×F2 )+ (K ×H). Since F1 × F2 is upper slightly I-continuous multifunction, it follows that there exis ...
... Proof. Let x ∈ X, K ⊂ Y and H ⊂ Z be clopen sets such that x ∈ F1+ (K) and x ∈ F2+ (H). Then we obtain that F1 (x) ⊂ K and F2 (x) ⊂ H and thus, F1 (x) × F2 (x) = (F1 ×F2 )(x) ⊂ K ×H. We have x ∈ (F1 ×F2 )+ (K ×H). Since F1 × F2 is upper slightly I-continuous multifunction, it follows that there exis ...
Convergence Measure Spaces
... x and its elements neighbourhoods of x. A set U ⊂ X is open if it is neighbourhood of each of its points. For each A ∈ X the adherence of A is the set a(A) = {x ∈ X : there is F ∈ λ(x) such that A ∈ F} and A ⊂ X is closed if a(A) = A. Remarks In general adherence operator need not be idempotent. Nei ...
... x and its elements neighbourhoods of x. A set U ⊂ X is open if it is neighbourhood of each of its points. For each A ∈ X the adherence of A is the set a(A) = {x ∈ X : there is F ∈ λ(x) such that A ∈ F} and A ⊂ X is closed if a(A) = A. Remarks In general adherence operator need not be idempotent. Nei ...
On the Decomposition of δ -β-I-open Set and Continuity in the Ideal
... where Cl(A) and Int(A) point out the closure and the interior of A, respectively. In [6], a point x ∈ X is called a δ-cluster point of A if A ∩ V = ∅ for every regular open set V containing x. The set of all δ-cluster point of A is called the δ-closure of A and denoted by Clδ (A). If Clδ (A) = A, t ...
... where Cl(A) and Int(A) point out the closure and the interior of A, respectively. In [6], a point x ∈ X is called a δ-cluster point of A if A ∩ V = ∅ for every regular open set V containing x. The set of all δ-cluster point of A is called the δ-closure of A and denoted by Clδ (A). If Clδ (A) = A, t ...
