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

1. 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),
2. Fréchetness of products (S. Dolecki and T. Nogura. Two-fold theorem on Fréchetness of products. Czech. Math. J., 49 (124):421-429, 1999),
3. Almost countably productive sequential properties (S. Dolecki and T. Nogura. Sequentially compact \alpha₃ sequential spaces of given order are almost countably productive. to appear),
4. Characterizations of subsequential topologies (S. Dolecki and S. Watson. Internal characterizations of subsequential topologies. to appear),
5. 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