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3 Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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On the almost uniform convergence

Drozdowski R., Jędrzejewski J., Sochaczewska A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2013

Vol. 18

11--17

Uniform convergence for continuous real functions sequences preserves continuity of the limit of such sequences. There are weaker types of convergence which have similar properties. We consider such types of convergence for functions from one topological space into another one.

On upper and lower strong quasi-uniform convergence of multivalued maps

Dominik I., Sochaczewska A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2012

Vol. 17

25-35

The concept of strong convergence of functions and multifunctions was introduced by I. Kupka, V. Toma and A. Sochaczewska. In this paper we consider new deﬁnitions of convergence for the nets of multifunctions – upper and lower strong quasi-uniform convergence.

On the quasi-uniform convergence

Drozdowski R., Jędrzejewski J., Sochaczewska A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2011

Vol. 16

19--22

Arzelá [1] considered the weaker form of uniform convergence which is as good as uniform convergence of sequences of functions in respect to continuity of the limit of a sequence of continuous functions. Some generalization of such convergence can be found in [5]. Similar kinds of convergence of function sequences were considered in [3] and [4]. In our article we generalize those kinds of convergence for functions defined in a topological space with values in a topological space. In the article we use terminology which is explained in Engelking's monograph “General Topology” [2]. Among others, we use the notion of a star with respect to an open over. If X is a topological space and α is a cover of this space, then the star St(x, α) of a point x ϵ X with respect to the cover α is defined as the union of all the sets from α which contain the point x, i.e. [wzór].