On the quasi-uniform convergence

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].