Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

3
The categories of this paper are: physics
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th