Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

In the paper we apply some of the results from the theory of ball spaces in the semimetric spaces. This allowed us to obtain some fixed point theorems which we believe to be unknown to this day. We also show the limitations of the ball space approach to this topic. As a byproduct, we obtain the equivalence of some different notions of completness in semimetric spaces where the distance function is $1$-continuous. In the second part of the article, we generalize Caristi-Kirk results for for $b$-metric spaces. Additionally, we obtain characterization of semicompleteness for $1$-continuous $b$-metric space via fixed point theorem analogous to the result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.