Seminarium 01.06.2023

**Properties of K-additive set-valued maps**

(joint work with Wojciech Jabłoński)

(joint work with Wojciech Jabłoński)

*Eliza Jabłońska*

(Akademia Górniczo Hutnicza)

(Akademia Górniczo Hutnicza)

We show selected properties of K-additive set-valued maps. First, we present a connection between K-additive s.v. maps and K-Jensen s.v. maps. Next, we prove that in the class of K-additive s.v. maps weak K-upper boundedness as well as K-lower boundedness on a "large" set imply K-continuity on the whole domain and, moreover, K-continuity implies K-homogeneity. Finally, we show that under some additional assumptions K-homogeneity implies K-continuity of a K-additive s.v. map. If time permits, we study an algebraic structure of the K-homogeneity set of a K-additive s.v. map.

Seminarium 15.06.2023

**Dependence on countable many of coordinates of continuous and separately continuous functions**

*Volodymyr Mykhailiuk*

We introduce some cardinal function on the product $X_1\times \cdots \times X_n$ of topological spaces $X_k$, which are formulated in terms of the type of local finiteness of families of open sets. Using this cardinal function, we obtain necessary and sufficient conditions that every separately continuous function $f:X_1\times\cdots \times X_n\to \mathbb{R}$ depends on countable many coordinates, where every space $X_k$ is a strongly countably Cech complete space.

Seminarium 22.06.2023

*Egmont Porten*

(Mid Sweden University)

(Mid Sweden University)