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Dissertation angenommen durch: Universität Rostock,
Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik,
2002-08-30
BetreuerIn: Prof. Dr. Harry Poppe , Universität Rostock, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik
GutachterIn: Prof. Dr. Gerhard Preuß , Freie Universität Berlin, Fachbereich Mathematik und Informatik
GutachterIn: Prof.Dr. Som Naimpally , Lakehead University, Thunder Bay, Canada, Dept. of Math.
Schlüsselwörter in Englisch: compactness, Ascoli, hyperspace, function space, precompactness, uniform, multifilter
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Abstrakt in Englisch
'Compactness properties of function space topologies, usually referred
to as Ascoli-Arzela type results, are most fundamental and useful in
both Topology and Analysis.' (S.A. Naimpally) Here we contribute new
aspects and strong new results to this widely studied topic through an
unusual approach involving hyperspaces. We consider topological spaces
and set-open topologies, as well, as we study a generalization of
Tukey's approach to uniformity, namely the strong topological universe
of multifilter spaces and fine maps, which may be viewed as the
extension of the classical (and not unsubstantiated) dichotomy in
descriptions of uniform structures into the realm of 'convenient
topology', developed by Gerhard Preuß. Hyperspaces are studied for
topological spaces as well as for multifilter spaces. Mostly emphasized
are compactness properties for hit-and-miss topologies from topological
spaces, simply, because they form the model for our new approach to
Ascoli-theorems in this work. Nevertheless, not all results are
completely devoted to this attempt - we think, some could be
interesting in their own right. There is a fairly useful
set-theoretical lemma, for instance, and a property called 'weak
relative complete' is considered for subsets of topological spaces. It
is a common generalization of closedness and compactness, and in fact
it is exactly what is needed to get compactness from relative
compactness. It is proved, that a hit-and-miss hyperspace, containing
at least the nonempty closed subsets, is compact if and only if the
base space is, whenever the miss-sets come from weak relative complete
subsets. Most of the former known compactness results for Fell or
Vietoris topology follow easily from this. Furthermore, a few results
on (relative) compactness of unions of (relative) compact subsets are
established. Concerning hyperstructures from multifilter-spaces, we
feel a quite direct transcription of the Vietoris-construction being
fruitful and we give a lemma concerning precompactness of unions of
precompact sets here. Finally, we devote the main part of the text to
the study of the simple but important map, as provided in a recent
(1998) embedding theorem of Mizokami. The topological behaviour of this
map is the key tool, that allow us to use our knowledge on hyperspaces
to produce powerful theorems of the Ascoli-Arzela type.
Bemerkungen in Deutsch
Erschien als Rostock, Univ., Diss., 2002 beim Shaker-Verlag, Aachen 2002. - ISBN 3-8322-0771-6
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