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Galway 4: The Fourth Galway Topology Colloquium at Birmingham
September 11-13, 2000
The University of Birmingham
Birmingham, UK |
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Organizers Chris Good (University of Birmingham)
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Technique of multifilters
by
Szymon Dolecki
Burgundy University, Dijon (France)
A cascade is a tree with a least element \varnothing well-founded
for the inverse order and such that its each (non maximal) element is a
filter on the set of its immediate successors. A multifilter on A
is a map from the set of maximal elements of a cascade to A. A multifilter
\Phi:maxT --> A converges to x in X if there is an
extension \Psi:T --> X of \Phi such that for every t in T\maxT, \Psi(t) in lim\Psi(t\natural ) where \Psi(t\natural ) stands for the image by \Psi of the filter t, while \Psi(t) is the image by \Psi of the point t. The contour is a
filter defined by induction by a diagonalizing process (S. Dolecki and
F. Mynard. Cascades and multifilters. Topology Appl., 104:53-65, 2000).
Sequential cascades are the cascades of countable rank each non maximal
element of which is a free sequential filter. A multifilter from a
sequential cascade is called a multisequence (S. Dolecki and
S. Sitou. Sur l'ordre séquentiel du produit de deux espaces de Fréchet. C.R.Acad.Sc. Paris, 322:465-470, 1996).
Multifilters are designed to describe the action of iterated adherences.
They apply in sundry situations, like
- Sequentiality and sequential order of products
(S. Dolecki and
S. Sitou. Precise bounds for sequential order of products of some Fréchet topologies. Topology Appl., 84:61-75, 1998, S. Dolecki
and T. Nogura. Sequential order of finite products of topologies. to appear),
- Fréchetness of products
(S. Dolecki and T. Nogura. Two-fold
theorem on Fréchetness of products. Czech. Math. J., 49 (124):421-429, 1999),
- Almost countably productive sequential properties
(S. Dolecki and
T. Nogura. Sequentially compact \alpha3 sequential spaces of given
order are almost countably productive. to appear),
- Characterizations of subsequential topologies
(S. Dolecki and
S. Watson. Internal characterizations of subsequential topologies. to
appear),
- Quotientness of product maps
(S. Dolecki and F. Mynard.
Convergence-theoretic mechanisms behind product theorems. Topology
Appl., 104:67-99, 2000).
Date received: September 2, 2000
Copyright © 2000 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Mathematical Conference Abstracts.
Document # cafq-10.