Frédéric Mynard

Convergence-theoretic mechanisms behind product theorems

Abstract Commutation of the topologizer with products, quotientness of product maps, preservation of some properties by products, topologicity of continuous convergence, continuity of complete lattices are facets of the same quest. A new method of multifilters is used to establish (in terms of core-contour-compactness) sufficient and necessary conditions for these properties in the framework of general convergences. The relativized Antoine reflector plays here an important role. Several classical results (of Whitehead, Michael, Boehme, Cohen, Day and Kelly, Hofmann and Lawson, Schwarz and Weck, Kent and Richardson, and others) are extended or refined.

Function space topologies

This chapter provides an overview of function space topologies. If Y and Z are two fixed topological spaces, C ( Y , Z ) denotes the set of all continuous maps from Y to Z , and t is a topology on the set C ( Y , Z ), then the corresponding topological space is denoted by C t ( Y , Z ). Each topology t on a set X defines a topological convergence class, denoted by C ( t ), consisting of all pairs ( F , s ) where F is a filter on X converging topologically to s ∈ X . A filter F on C ( Y , Z ) converges continuously to a function f if e ( F × G ) (where e : C ( Y , Z ) × Y → Z is the evaluation map) converges to f(x ) in Z whenever G is a filter convergent to x in X . A subset B of a space X is called bounded or relatively compact if every open cover of X contains a finite sub-cover of B . The concepts related to splitting and admissible topologies are presented. Details of the greatest splitting, compact-open, and Isbell topologies are also provided in the chapter.

Compatible relations on filters and stability of local topological properties under supremum and product

Abstract An abstract scheme using particular types of relations on filters leads to general unifying results on stability under supremum and product of local topological properties. We present applications for Frechetness, strong Frechetness, countable tightness and countable fan-tightness, some of which recover or refine classical results, some of which are new. The reader may find other applications as well.

Cascades and multifilters

Abstract Cascades (trees every element of which is a filter on the set of its successors), and multifilters, maps from cascades, are introduced. Multisequences constitute a special case of multifilters. Applications to convergence and to topology are indicated.

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.

Espaces productivement de Fréchet

Resume Les espaces topologiques dont le produit avec chaque espace fortement de Frechet est de Frechet sont caracterises de facon interne. Ceci resout un probleme reste longtemps ouvert. Pour citer cet article : F. Jordan, F. Mynard, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 259–262.

Products of compact filters and applications to classical product theorems

Abstract Two results on product of compact filters are shown to be the common principle behind a surprisingly large number of theorems.

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.

Measure of Compactness for Filters in the Approach Setting

New explicit descriptions of the reflections of a convergence-approach space on pseudo-approach, pre-approach and approach spaces are given. A measure of compactness for filters is introduced in the context of convergence-approach spaces. It is shown that this measure generalizes all the known measures of variants of compactness and can also be used to describe the reflections mentioned above.

Distinguishing the Plane from the Punctured Plane Without Homotopy

This is a note about topology, meant to illustrate a specific technique. If you do not know what topology is about, think of it this way: if you look at two congruent triangles in the plane, you co...