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Contents:Continuity-like Properties of Mappings (PhD Thesis, MFF UK 1997)
Preface, Introduction, the introductory notes to the remaining chapters and Bibliography are contained in this pdf file (297 kB). Except for this the thesis contains six published papers from the above list.
- Valdivia compacta and continuous images
- Markushevich bases and primarily Lindelöf spaces
- Valdivia type Banach spaces
- Introduction - pdf file (58 kB)
- Remark on the Point of Continuity Property II (a published joint paper with P.Holicky, see above)
- Note on Connections of Point of Continuity Property and Kuratowski Problem on Function Having the Baire Property (a published paper, see above)
- New Examples of Hereditarily t-Baire Spaces (a published paper, see above)
- Stegall Compact Spaces Which Are Not Fragmentable (a published paper, see above)
- Few remarks on structure of certain spaces of measures - pdf file (209 kB)
We give an example of an Asplund space - with an equivalent norm - whose bidual unit ball is not a Valdivia compact. This answers in the negative a question of J.Orihuela (1992). Further we introduce a subclass of Asplund spaces, which has nice stability properties and each element of which has Valdivia bidual unit ball. This class contains all Asplund spaces whose dual unit ball is -Corson (in particular, all Asplund spaces of density and all Asplund weakly compactly generated spaces).
We prove in particular that the dual unit ball of a Banach space X is a Corson compact provided X is of the form C(K) where K is a continuous image of a Valdivia compact space, and the dual unit ball of every subspace of X is Valdivia compact. Another result is that, if K is a non-Corson continuous image of a Valdivia compact, there is a continuous image L ofK such that the dual unit ball of C(L) is not Valdivia.
We prove in particular that a continuous image of a Valdivia compact space is Corson provided it contains no homeomorphic copy of the ordinal segment . This generalizes a result of R.Deville and G.Godefroy who proved it for Valdivia compact spaces. We give also a refinement of their result which yields a pointwise version of retractions on a Valdivia compact space.
We prove that the dual unit ball of a Banach space Xis a Corson compact provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compact. As a corollary we get that the dual unit ball of a Banach space X of density is Corson if X has projectional resolution of the identity with respect to every equivalent norm. These results answer questions posed by M.Fabian, G.Godefroy and V.Zizler.
We characterize Valdivia compact spaces K in terms of C(K) endowed with a topology introduced by M.Valdivia (1991). This generalizes R.Pol's characterization of Corson compact spaces. Further we study duality, products and open continuous images of Valdivia compact spaces. We prove in particular that the dual unit ball of C(K) is Valdivia wheneverK is Valdivia and that the converse holds whenever K has a dense set of points. Another result is that any open continuous image of a Valdivia compact space with a dense set of points is again Valdivia.
We study topological properties of Valdivia compact spaces. We prove in particular that a compact Hausdorff space K is Corson provided each continuous image of K is a Valdivia compact. This answers a question of M.Valdivia (1997). Another results are that the class of Valdivia compacta is stable with respect to arbitrary products and a generalization of the fact that Corson compacta are angelic.
We prove that the dual unit ball of the space endowed with the weak* topology is not a Valdivia compact. This answers a question posed to the author by V.Zizler and has several consequences. Namely, it yields an example of an affine continuous image of a convex Valdivia compact (in the weak* topology of a dual Banach space) which is not Valdivia, and shows that the property of the dual unit ball being Valdivia is not an isomorphic property. Another consequence is that the space has no countably 1-norming Markusevic basis.
The spaces of Borel probabilities on a topological space X inherit a number of topological properties of X. We show in particular that the space of tight probabilities on a Cech-analytic space is Cech-analytic. Analogical results are shown for several other classes of generalized analytic and complete topological spaces.
Using modifications of the well-known construction of "double-arrow" space we give consistent examples of nonfragmentable compact Hausdorff spaces which belong to Stegall's class . Namely the following is proved.
(1) If is less than the least inaccessible cardinal in and hold then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a Baire space intoK is singlevalued at points of a residual set.
(2) If V=L then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a completely regular Baire space intoK is singlevalued at points of a residual set.
We introduce a new class of hereditarily t-Baire spaces (defined by G.Koumoullis (1993)) which need not to have the restricted Baire property in a compactification - as an example serves the space for A uncountable. We use this and a modification of a construction of D.Fremlin (1987) to get, under the assumption that there is a measurable cardinal, an example of a first class function of a hereditarily t-Baire space into a metric space which has no point of continuity, which shows, in answer to a question of G.Koumoullis (1993), that the cardinality restriction in his Theorem 4.1 cannot be dropped.
It is shown in particular that the question whether every extended Borel class one (e.g. -measurable) map of any hereditarily Baire space into a metric space has the point of continuity property is equivalent to the Kuratowski question whether the function with the Baire property of any topological space into a metric space is continuous apart from a meager set. The method of the proof enables us to get, under the assumption that it is consistent to suppose that there is a measurable cardinal, examples of ordinary Borel class one maps (i.e. -measurable) of a hereditarily Baire space intoa metric space which have not the point of continuity property.These examples complete and strengthen an example of G.Koumoullis, who constructed (under the assumption that there is a real-valued measurable cardinal an extended Borel class one function (even -measurable) of a hereditarily Baire space into a metric space with no continuity point but it is not clear whether this map is -measurable.
We prove in particular that if X is a hereditarily Baire space which has the tightness less than the least weakly inaccessible cardinal and each (closed) subspace of Xhas the countable chain condition, then every Borel class one map of X into a metric space M has the point of continuity property. In the case of countable tightness the assumption that every closed subspace has the countable chain condition is not needed and we get a result of R.W.Hansell (1991).