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Frédéric Mynard

Publications and source records attributed to Frédéric Mynard.

11 recordsLinked to original sources
arXiv2025

On (pre-) approach spaces within convergence approach spaces

The purpose of this note is to illustrate a parallel between (pre)topologies when seen among convergence spaces and (pre)approach spaces when seen among convergence approach spaces, that appears to be a more complete parallel than in the traditional presentation of these approach structures. This sheds some light on the reflector from the category of convergence approach spaces to that of approach spaces and even on the structure of approach spaces as such. This point of view allows for a characterization of approach spaces among convergence approach spaces represented as pointfree convergence frames as in [18].

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arXiv2025

On connected subsets of a convergence space

Though a convergence space is connected if and only if its topological modification is connected, connected subsets differ for the convergence and for its topological modification. We explore for what subsets connectedness for the convergence or for the topological modification turn out to be equivalent. In particular, we show that the largest set containing a given connected set for which all subsets in between are connected is the adherence of the connected set for the reciprocal modification of the convergence.

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arXiv2021

When is a category of adherence-determined convergences simple?

Abstract. We provide a characterization of classes of filters $\mathbb{D}$ for which the full subcategory $\operatorname{fix}\operatorname{A}_{\mathbb D}$ of $\mathsf{Conv}$ formed by convergences determined by the adherence of filters of the class $\mathbb{D}$ is simple in $\mathsf{Conv}$. Along the way, we also elucidate when two classes of filters result in the same category of adherence-determined convergences. As an application of the main result, we show that the category of hypotopologies is not simple, thus answering a question from [25].

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arXiv2018

On (ultra-) completeness numbers and (pseudo-) paving numbers

We study the completeness and ultracompleteness numbers of a convergence space. In the case of a completely regular topological space, the completeness number is countable if and only if the space is $Č$ech-complete, and the ultracompleteness number is countable if and only if the space is ultracomplete. We show that the completeness number of a space is equal to the pseudopaving number of the upper Kuratowski convergence on the space of its closed subsets, at $\emptyset$. Similarly, the ultracocompleteness number of a space is equal to the paving number of the upper Kuratowski convergence on the space of its closed subsets, at $\emptyset$.

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arXiv2018

Convergence without points

We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending every filter of subsets to its set of limits. This construction exhibits the category of convergence spaces as a coreflective subcategory of the opposite of the category of convergence lattices. We extend this construction to coreflections between limit spaces and the opposite of so-called limit lattices and limit coframes, between pretopological convergence spaces and the opposite of so-called pretopological convergence coframes, between adherence spaces and the opposite of so-called adherence coframes, between topological spaces and the opposite of so-called topological coframes. All of our pointfree categories are complete and cocomplete, and topological over the category of coframes. Our final pointfree category, that of topological coframes, shares with the category of frames the property of being in a dual adjunction with the category of topological spaces. We show that the latter arises as a retract of the former, and that this retraction restricts to a reflection between frames and so-called strong topological coframes.

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arXiv2018

The convergence-theoretic approach to g-first countable and symmetrizable spaces

This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Fréchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

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arXiv2014

Measure of compactness for filters in product spaces: Kuratowski-Mròwka in CAP revisited

The first author introduced a measure of compactness for families of sets, relative to a class of filters, in the context of convergence approach spaces. We characterize a variety of maps (types of quotient maps, closed maps, and variants of perfect maps) as those respecting this measure of compactness under one form or another. We establish a product theorem for measure of compactness that yields as instances new product theorems for spaces and maps, and new product characterizations of spaces and maps, thus extending existing results from the category of convergence spaces to that of convergence approach spaces. In particular, results of the Mròwka-Kuratowski type are obtained, shedding new light on existing results for approach spaces.

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arXiv2014

Function spaces and contractive extensions in Approach Theory: The role of regularity

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.

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arXiv2013

Exponential Kleisli monoids as Eilenberg-Moore algebras

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

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