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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

##### MSC:
 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