0000000000539947
AUTHOR
Frédéric Mynard
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.
Group topologies coarser than the Isbell topology
Abstract The Isbell, compact-open and point-open topologies on the set C ( X , 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 , R ) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α ( X ) for which C α ( X , 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, t…
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.