Applications of covergence theory in general topology
by
Szymon Dolecki
Université de Bourgogne

A convergence on X is an isotone relation between the filters on X and the points of X. Topologies, pretopologies, paratopologies and pseudotopologies are important supremum-closed classes of convergences; sequence, first-countable and locally compact convergences are examples of convergence classes closed for infima.

It is known that such topology classes like sequential, Fréchet, strongly Fréchet, k, k', strongly k', bi-sequential can be characterized as the solutions of the inequalities of the type

\tau = 84 J E \tau, \label1

where J is a projection on a convergence class closed for suprema while E is a coprojection on a convergence class closed for infima. For instance, a topology \tau is Fréchet iff holds for J the pretopologization and E the sequentialization.

It was recently observed that classical types of quotient maps f (from \xi to \tau) can be characterized by

\tau = 84 J(f \xi), \label2

where f \xi stands for the final covergence of \xi with respect to f and J is a projection (for example, f is almost open iff J is the identity, f is bi-quotient iff J is the pseudotopologization, f is countably bi-quotient iff J is the paratopologization, f is hereditarily quotient iff J is the pretopologization, f is quotient iff J is the topologization).

These two characterization schemes imply a preservation/reconstruction scheme that include numerous classical results:

\tau = 84 J E \tau iff \tau is the image by f with of a convergence \xi fulfilling \xi = 84 J E \xi.

It turns out that covering maps (like sequence-covering or compact-covering) can be represented as certain quotient maps with respect to some coprojections E of the original convergences. Namely,

E \tau = 84 J f(E \xi)

holds for J the identity and E the sequentialization iff f is sequence-covering; for J the pseudotopologization and E the compact localization provided that f is compact-covering. Therefore such classical facts like``sequence-covering map on a Fréchet topology is hereditarily quotient'' are immediate consequences of

If and , then .

Accessibility and strong accessibility spaces (, ) turn out to be special classes of J-maximal convergences with respectto D, where J and D are both projections. Accordingly, the following theorem recovers, as special cases, several classical facts:

If \tau is J-maximal with respect to D and holds, then \tau = 84 D(f\xi).

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S. Dolecki. Conevrgence-theoretic approach to quotient quest. Topology Appl. to appear, 1996.

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S. Dolecki and G. H. Greco. Topologically maximal pretopologies. Studia Math., 77:265-281, 1984.

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S. Dolecki and G. H. Greco. Cyrtologies of convergences, II: Sequential convergences. Math. Nachr., 127:317-334, 1986.

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S. Dolecki and M. Pillot. Topologically maximal convergences and types of accessibility. to appear.

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F. Siwiec. Sequence-covering and countably bi-quotient mappings. Gen. Topology Appl., 1:143-154, 1971.

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G. T. Whyburn. Accessibility spaces. Proc. Amer. Math. Soc., 24:181-185, 1970.

Date received: June 24, 1996

Copyright © 1996 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 Conferences Inc. Document # caaj-34.