On some convergence approach structures on hyperspaces

In the context of the category $\mathsf{Cap}$ of convergence approach spaces and contractions, we introduce and study approach analogs of the upper and lower Kuratowski convergences, upper-Fell and Fell topologies on the set of closed subsets of the coreflection on the category $\mathsf{Conv}$ of convergence spaces of a convergence approach space. In particular, over a pre-approach space, the $\mathsf{Conv}$-coreflection of the lower Kuratowski convergence approach structure is the lower Kuratowski convergence associated with the $\mathsf{Conv}$-coreflection of the base space, while the $\mathsf{Conv}$-reflection is the lower Kuratowski convergence associated with the $\mathsf{Conv}$-reflection. The $\mathsf{Conv}$-coreflection of the upper Kuratowski convergence approach is is the upper Kuratowski convergence associated with the $\mathsf{Conv}$-reflection of the base space, while the $\mathsf{Conv}$-reflection is the upper Kuratowski convergence associated with the $\mathsf{Conv}$-coreflection of the base space. We show that, over an approach space, the lower Kuratowski convergence approach structure is in fact an approach structure that coincides with the $\vee$-Vietoris approach structure introduced by Lowen and his collaborators, though it may be strictly finer over a general convergence approach space. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach, but finer than the upper Fell approach structure introduced by the first and third author. We also obtain a $\mathsf{Cap}$ abstraction of the classical result that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.

💡 Research Summary

The paper works within the category Cap of convergence‑approach spaces (objects equipped with an approach‐distance λ on filters) and contractions (maps preserving λ). It studies hyperspaces, i.e., the set C X of closed subsets of a given space X, and introduces approach‑theoretic analogues of the classical Kuratowski upper and lower convergences as well as the Fell topologies.

First, the authors recall the coreflection Conv‑coref and reflection Conv‑ref from Cap to the category Conv of ordinary convergence spaces. They show how these functors act on a base space and on its hyperspace. For a pre‑approach space (X,λ) they prove that the Conv‑coreflection of the lower Kuratowski approach structure on C X coincides with the lower Kuratowski convergence on C X derived from the Conv‑coreflection of X. Dually, the Conv‑reflection of the upper Kuratowski approach structure coincides with the upper Kuratowski convergence on C X derived from the Conv‑reflection of X. Thus the coreflection “preserves” the lower Kuratowski side, while the reflection “preserves” the upper side.

Next, the paper analyses the intrinsic nature of these hyperspace approach structures. The lower Kuratowski approach on C X is shown to be exactly the ∨‑Vietoris approach structure introduced by Lowen et al.; consequently, on an approach space it is an approach space (i.e., λ satisfies the finite‑depth axiom) and coincides with the classical lower Vietoris topology. However, for a general convergence‑approach space the ∨‑Vietoris structure can be strictly coarser, indicating that the lower Kuratowski approach may carry more refined convergence information.

The upper Kuratowski approach is always pseudo‑topological. The authors then define an “upper Fell” approach structure, which is non‑Archimedean (the distance between two filters is the supremum of distances of their members). They prove that this upper Fell structure is coarser than the upper Kuratowski approach but finer than the previously introduced upper Fell approach (by the first and third authors). Hence we obtain a strict chain: upper Fell (new) ⊂ upper Kuratowski ⊂ upper Fell (old).

A significant part of the work is devoted to relating these approach structures to classical topological results. The authors abstract the well‑known theorem: if the upper Kuratowski convergence on a topological space is pretopological, then it is already topological. In the Cap setting they show that pretopologicity of the upper Kuratowski approach forces its Conv‑coreflection and Conv‑reflection to coincide, which yields a genuine topology on C X. This provides a categorical version of the classical result.

Throughout the paper, detailed calculations with filters, reductions (rdc), and the isotone hulls are carried out. Lemmas concerning saturated filters, the behavior of the reduction operator under the Fell subbasis, and the interaction between nets and filters are used to bridge the net‑based classical definitions with the filter‑based approach framework.

In summary, the contributions are:

1. Precise description of how Conv‑coreflection and Conv‑reflection transform lower and upper Kuratowski approach structures on hyperspaces.
2. Identification of the lower Kuratowski approach with the ∨‑Vietoris approach, establishing its status as an approach space and clarifying its relation to the lower Vietoris topology.
3. Introduction of a new non‑Archimedean upper Fell approach, positioned strictly between the upper Kuratowski and the earlier upper Fell approaches.
4. A Cap‑level abstraction of the classical pretopological ⇒ topological theorem for upper Kuratowski convergence.

These results enrich the theory of hyperspaces by embedding classical convergence and topological notions into the more flexible framework of convergence‑approach spaces, opening avenues for further exploration of metric‑like structures on hyperspaces, non‑symmetric topologies, and categorical reflections in related categories.