BazTech - Yadda

1

A criterion for local resolvability of a space and the ω-problem

Ponomarev S. P.

Journal of Applied Analysis

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2007

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Vol. 13, nr 1

83-96

EN

The notion of the resolvability of a topological space was introduced by E. Hewitt [8]. Recently it was understood that this notion is also important in the study of ω)-primitives, especially in the case of nonmetrizable spaces. In the present paper a criterion for the resolvability of a topological space at a point ("local resolvability") is given. This criterion, stated in terms of oscillation and quasicontinuity, permits to conclude, for instance, that on irresolvable spaces no positive continuous real-valued function has an ω-primitive. The result is strenghtened in the case of SI-spaces. It is also shown that every non- negative upper semicontinuous function on a resolvable Baire space has an ω)-primitive.

2

On the integral quasicontinuity

Grande Z., Strońska E.

Journal of Applied Analysis

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2006

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Vol. 12, nr 2

233-248

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.

3

A note on the quasicontinuity

Grande Z.

Demonstratio Mathematica

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2006

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Vol. 39, nr 3

515-518

EN

Let (X,px) and (V, pv) be metric spaces. A function f: X - Y is said graph quasicontinuous if there is a quasicontinuous function g : X -> Y with the graph Gr(g) contained in the closure cl(Gr(f)) of Gr(f). If the space (V, pv) is compact and if there is a dense subset A C X such that the restricted function f/A is continuous then f is graph quasicontinuous. Moreover each locally bounded function f: R -> R is graph quasicontinuous.

4

On universal elements for some families of functions

Grande Z., Strońska E.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

679--688

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.

5

On cliquishness of maps of two variables

Ewart J.

Demonstratio Mathematica

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2002

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Vol. 35, nr 3

657-670

EN

We give some sufficient conditions under which a separately cliquish map of two variables is cliquish.

6

On sequences of upper and lower semi-quasicontinuous functions

Grande Z.

Journal of Applied Analysis

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2001

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Vol. 7, nr 2

235-242

EN

In this article we investigate the pointwise, discrete and transfinite convergences in the classes of real functions defined on topological spaces which are upper and lower quasicontinuous at each point.

7

On some representations of functions discontinuous on countable sets

Błaszczyk T., Grande Z.

Demonstratio Mathematica

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2001

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Vol. 34, nr 3

543-548

EN

It is proved that every function f : R -> R having countably many of discontinuity points is the sum of two bilaterally quasi-continuous functions which are continuous at every continuity point of f.