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1
On E-continuity and E-minimality of multifunctions of two variables
EN
In this article we formulate and prove some sufficient conditions for the l-Ex xy-continuity and the Ex x y- minimality of multifunctions of two variables.
EN
A function f : R w indeksie górnym m → R satisfies the condition [wzór] (resp. [wzór]) at a point x ∈ R w indeksie górnym m if for each real ε > 0 and for each set U ∋ x belongong to Euclidean topology in R w indeksie górnym (resp. to the strong density topolgy [to the ordinary density topology]) there is an open set 0 such that 0 ∩ U ≠ Ø and [wzór]. These notions are some analogies are some analogies of the quasicontinuity or the approximate quasicontinuity. In this article we compare these notions with the classical notion of the quasicontinuity.
EN
Let (X, Tx) be a topological space and let (Y, dy) be a metric space. For a function f : X → y denote by C(f) the set of all continuity points of f and by D(f) = X\C(f) the set of all discontinuity points of f. Let C(X,Y) = {f : X → Y; f is continuous}, H(X, Y) = {f: X →Y; D{f) is countable}, H1(X, Y) = {f: X → Y; ∃h ∈c(x,Y) {x; f(x) ≠ h{x)} is countable}, and H2(X, Y) = H(X, Y) ∩ H1(X, Y). In this article we investigate some convergences (pointwise, uniform, quasiuniform, discrete and transfinite) of sequences of functions from H(X, Y), H1(X, Y) and H2(X, Y).
4
On some special notions of approximate quasicontinuity on Rm
EN
Some special notions of approximate quasicontinuity on Rm and the uniform, pointwise, transfinite and the discrete convergence of sequences of such functions are investigated.
5
On universal elements for some families of functions
EN
A point x C X is called universal element for a family phi of functions from X to y if the set {f(x)\f 6 phi} is dense in Y. In this article we show that every residual G- set in a completely regular space X (every residual set in R ) is the set of all universal elements for some family of continuous functions from X to R (for some family of quasicontinuous functions from Rk to R). Moreover we investigate the sets of all universal elements for some families of monotone functions and for some families of functions having the property of Denjoy-Clarkson.
6
On an ideal of linear sets
EN
In this article we investigate the ideal of linear sets A such that for each nonempty set U contained in the closure cl(A) of A and belonging to the density topology the intersection U D A is a nowhere dense subset of U.
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