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F. Mynard

Publications and source records attributed to F. Mynard.

7 recordsLinked to original sources
arXiv2026

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.

math.GN↗
arXiv2022

On points of convergence lattices and sobriety for convergence spaces

We characterize the convergence spaces $(X,\xi)$ such that the space of points of $(\mathbb{P}X,\lim_{\xi})$ in the category of convergence lattices is $(X,\xi)$. On the way, we study variants of sobriety and of the axiom $T_{D}$ in convergence spaces. New phenomena appear when leaving the realm of topological spaces. We obtain new hindsight into the space of points of a convergence lattice and study a special quotient of it, which, in the case $L=(\mathbb{P}X,\lim_{\xi})$ for a topological space $(X,\xi)$, turns out to be homeomorphic to the sobrification of $X$.

math.GN↗
arXiv2010

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $α(X)$ of compact families of open subsets of a topological space $X$. Those $α(X)$ for which addition is jointly continuous at the zero function in $C_α(X,\mathbb{R})$ are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections $α(X)$ for which $C_α(X,\mathbb{R})$ is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if $X$ is infraconsonant. Examples based on measure theoretic methods, that $C_α(X,\mathbb{R})$ can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.

math.GN↗
arXiv2010

A unified theory of function spaces and hyperspaces: local properties

Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on $C(X,\mathbb R)$, such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{é}chet property and some of its variants. Under mild conditions, local properties of $C(X,\mathbb R)$ at the zero function correspond to the same property of the associated hyperspace structure at $X$. The latter is often easy to characterize in terms of covering properties of $X$. This way, many classical results are recovered or refined, and new results are obtained. In particular, it is shown that tightness and character coincide for the continuous convergence on $C(X,\mathbb R)$ and is equal to the Lindel{ö}f degree of $X$. As a consequence, if $X$ is consonant, the tightness of $C(X,\mathbb R)$ for the compact-open topology is equal to the Lindel{ö}f degree of $X$.

math.GN↗
arXiv2010

When is the {I}sbell topology a group topology?

Conditions on a topological space $X$ under which the space $C(X,\mathbb{R})$ of continuous real-valued maps with the Isbell topology $κ$ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in $C_κ(X,\mathbb{R})$ if and only if $X$ is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in $C_κ(X,\mathbb{R})$ if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if $X$ is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.

math.GN↗