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Algebraic structures generated by d-quasi continuous functions. (English) Zbl 0655.26007
Let (X,\({\mathcal T})\) be a topological space, R denotes the real line. A function \(f:\quad X\to R\) is said to be \({\mathcal T}\)-quasi continuous (\({\mathcal T}\)-cliquish) at a point \(x_ 0\in X\) iff for every \(\epsilon >0\) and for each neighborhood \(U\in {\mathcal T}\) of \(x_ 0\) there exists a \({\mathcal T}\)-open set V such that \(\emptyset \neq V\subset U\) and \(| f(x)-f(x_ 0)| <\epsilon\) for every \(x\in V\) \((| f(x_ 1)- f(x_ 2)| <\epsilon\) for \(x_ 1,\quad x_ 2\in V)\). A function f is \({\mathcal T}\)-quasi continuous (\({\mathcal T}\)-cliquish) on X iff f is \({\mathcal T}\)-quasi continuous (\({\mathcal T}\)-cliquish) at every point of X. Let d be the density topology on \(R^ m\). The authors prove that the algebra (lattice) generated by the family of d-quasi continuous functions and the collection of all pointwise limits of sequences of d-quasi continuous functions are equal to the family of d-cliquish functions.
Reviewer: P.Kostyrko

26B05 Continuity and differentiation questions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C05 Continuous maps