In a topological space, Kuratowski convergence of hypernets is defined by Beer[1] and in a convergence space, Kuratowski convergence of hyperfilters is defined by Dolecki and Mynard [5]. In this study, we introduce and study upper and lower Kuratowski convergences of hyperfilters in the category Cap of convergence approach spaces and contractions.

Given a convergence approach space $(X,\lambda)$, let $C_{c(\lambda)}$ denote $c(\lambda)$-closed subsets of $X$. For a hyperfilter $\mathfrak{F}$ defined on $C_{c(\lambda)}$ and $A\in C_{c(\lambda)}$ we defined:

$ \lambda_{uK}\mathfrak{F}(A)=\bigvee_{x\notin A}1\oslash adh_\lambda (rdc\mathfrak{F})(x)$,

$ \lambda_{lK}\mathfrak{F}(A)=\bigvee_{x\in A}1\oslash adh_\lambda (rdc\mathfrak{F}^\textit{#})(x)$

and

$ \lambda_{K}\mathfrak{F}(A)=\lambda_{uK}\mathfrak{F}(A) \bigvee \lambda_{lK}\mathfrak{F}(A). $

Given an $\epsilon\in[0,\infty]$, the filter $\mathfrak{F}$ is said to be $\epsilon-$upper Kuratowski convergent (respectively $\epsilon-$lower Kuratowski convergent, respectively $\epsilon-$ Kuratowski convergent) to $A$ if $\lambda_{uK}\mathfrak{F}(A)\leq\epsilon$ (respectively $\lambda_{lK}\mathfrak{F}(A)\leq\epsilon$, respectively $\lambda_{K}\mathfrak{F}(A)\leq\epsilon$).

We investigate the properties of this convergences and then obtain relations with these new notion of convergence and Fell approach structure defined by Ateş and Sagıroglu in [4]. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach structure, but finer than the upper Fell approach structure introduced in [4]. We also obtain that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.

References

[1] G. Beer, Topologies on Closed Convex Sets, Kluwer Academic Publishers, 1993.

[2] F. Mynard, Measure of compactness for filters in the approach setting, Quaestiones Math. 31:189-201, 2008.

[3] F. Mynard, On (pre)-approach spaces within convergence approach spaces, Topology Proceeding, 68: 41-66, 2026.

[4] M. Ates, S. Sagiroglu, The Fell Approach Structure, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 2023.

[5] S. Dolecki, F. Mynard, Convergence Foundations of Topology, World Scientific, 2016.

[6] R. Lowen, Approach Spaces: the Missing Link in the Topology Uniformity Metric Triad, Oxford Mathematical Monographs.
Oxford University Press, New York, United Stated Springer, 1997.

[7] R. Lowen, Index Analysis, Approach Theory at Work, Springer, 2015