Another note on Levine′s decomposition of continuity

Abstract

ABSTRACT. Several decompositions of continuity each stronger than Norman Levine's are found improving results of J. Chew and J. Tong, as well as of the first two named authors above.

Weak α(Λ, sp)-continuity for multifunctions

“…In 1987, Noiri [8] introduced the class of functions called weakly α-continuous functions. Popa and Noiri [11], Rose [14] and Sen and Bhattacharyya [15] studied some properties of weakly α-continuous functions. Several different forms of continuous multifunctions have been introduced and studied over the years.…”

(τ1,τ2)α-Continuity for Multifunctions

“…In 1987, Noiri [10] introduced a class of functions called weakly α-continuous functions. Some characterizations of weakly α-continuous functions are investigated in [11][12][13]. Neubrunn [14] extended these functions to multifunction and introduced the notions of upper and lower α-continuous multifunctions.…”

“…⟹ (12): Let V be any σ 1 σ 2 -open subset of Y. By(11) and Lemma 11 (1), we have (τ 1 , τ 2 )α-Cl(F -(σ 1 σ 2 -Int(σ 1 σ 2 -Cl(V)))) ⊆ F -(σ 1 σ 2 -Cl(V)). (12) ⟹ (10): Let V be any σ 1 σ 2 -open subset of Y. By(12), we haveτ 1 , τ 2 α-Cl Fσ 1 σ 2 -Int σ 1 σ 2 -Cl(V) ⊆ Fσ 1 σ 2 -Cl(V) , (48)and henceτ 1 , τ 2 α-Cl F -(V) ( ) ⊆ τ 1 , τ 2 α -Cl Fσ 1 σ 2 -Int σ 1 σ 2 -Cl(V) ⊆ Fσ 1 σ 2 -Cl(V). For a multifunction F: (X, τ 1 , τ 2 ) ⟶ (Y, σ 1 , σ 2 ), the following properties are equivalent:(1) F is lower weakly (τ 1 , τ 2 )α-continuous (2) For each x ∈ X and each σ 1 σ 2 -open subset V of Y such that F(x) ∩ V ≠ ∅, there exists a (τ 1 , τ 2 )α-open subset U of X containing x such that U ⊆ F − (σ 1 σ 2 -Cl(V)) (3) F − (V) ⊆ τ 1 -Int(τ 2 -Cl(τ 1 τ 2 -Int(F − (σ 1 σ 2 -Cl(V))))) for every σ 1 σ 2 -open subset V of Y (4) τ 1 -Cl(τ 2 -Int(τ 1 τ 2 -Cl(F + (σ 1 σ 2 -Int(K))))) ⊆ F + (K) for every σ 1 σ 2 -closed subset K of Y (5) (τ 1 , τ 2 )α-Cl(F + (σ 1 σ 2 -Int(K))) ⊆ F + (K) for every σ 1 σ 2closed subset K of Y (6) (τ 1 , τ 2 )α-Cl(F + (σ 1 σ 2 -Int(σ 1 σ 2 -Cl(B)))) ⊆ F + (σ 1 σ 2 -Cl(B)) for every subset B of Y (7) F − (σ 1 σ 2 -Int(B)) ⊆ (τ 1 , τ 2 )α-Int(F − (σ 1 σ 2 -Cl(σ 1 σ 2 -Int(B)))) for every subset B of Y (8) F − (V) ⊆ (τ 1 , τ 2 )α-Int(F − (σ 1 σ 2 -Cl(V))) for every σ 1 σ 2open subset V of Y (9) (τ 1 , τ 2 )α-Cl(F + (σ 1 σ 2 -Int(K))) ⊆ F + (K) for every (σ 1 , σ 2 )r-closed subset K of Y (10) (τ 1 , τ 2 )α-Cl(F + (V)) ⊆ F + (σ 1 σ 2 -Cl(V)) for every σ 1 σ 2open subset V of Y (11) (τ 1 , τ 2 )α-Cl(F + (σ 1 σ 2 -Int((σ 1 , σ 2 )θ-Cl(B)))) ⊆ F + ((σ 1 , σ 2 )θ-Cl(B)) for every subset B of Y (12) (τ 1 , τ 2 )α-Cl(F + (σ 1 σ 2 -Int(σ 1 σ 2 -Cl(V)))) ⊆ F + (σ 1 σ 2 -Cl(V)) for every σ 1 σ 2 -open subset V of YProof.…”

Continuity Points of Functions on Product Spaces

“…Many authors consider the notion of local w * continuity as a suitable tool for investigation of relationship between continuity and connectivity [3], [6], [7], [10]. There arises a question whether it is possible to define a point version of local w * continuity.…”