We provethat f has shadowing (h-shadowing) if and only if 2f has shadowing(h-shadowing).ĪB - Given a nonempty compact metric space X and a continuousfunction f : X → X, we study shadowing and h-shadowing forthe induced maps on hyperspaces, particularly in symmetric products,Fn(X), and the hyperspace of compact subsets of X, 2X. N2 - Given a nonempty compact metric space X and a continuousfunction f : X → X, we study shadowing and h-shadowing forthe induced maps on hyperspaces, particularly in symmetric products,Fn(X), and the hyperspace of compact subsets of X, 2X. Some relevant properties which insure H-singularity are introduced and investigated in some aspect.T1 - Shadowing for induced maps of hyperspaces This characterization of H-minimality becomes a criterion for H-singularity for all uniformities that are either complete, uniformly locally precompact or proximally fine (e.g., metrizable ones). nectedness, local connectedness and compactness of the hyperspace TD(X). Here we begin the study of the hyperspace. Besides, we provide an intrinsic characterization of H-minimal uniformities and show that they are H-coarse. a space through its hyperspaces and vice versa, see references in the classical books in this matter 5 and 10. We show, by assuming the existence of \\mathfrak c-many selective ultrafilters, that there exists a subspace of that is (, )-pseudocompact for all <\\mathfrak c, but \\text. We show here that there is a property which obstructs H-singularity: Every H-minimal uniformity has a base of finite-dimensional uniform coverings. We study the relations between a generalization of pseudocompactness, named (, M)-pseudocompactness, the countably compactness of subspaces of and the pseudocompactness of their hyperspaces. The self-explanatory concepts of H-coarse, H-minimal and H-maximal uniformities are defined similarly.It is well known that not. The uniformity U is said to be H-singular if no distinct uniformity on X is H-equivalent to U. Two uniformities U and V on a set X are said to be H-equivalent if their corresponding Hausdorff uniformities on the set of all non-empty subsets of X induce the same topology. Several remarks related to hyperspaces of countably compact (pseudocompact) spaces are given in the last section of the paper. Finally, as a natural step in order to generalize the above result, we study which kind of metric or uniform spaces satisfy that the Samuel realcompactification and the Hewitt realcompactification are equivalent. compact space X to guarantee the countable compactness of 2X. In another section of the paper, we have studied the local compactness of hyperspaces, endowed with either the Vietoris, the Wijsman, or the Hausdorff. It turns out that these are exactly the Bourbaki-complete uniform spaces having no uniform partition of Ulam-measurable cardinal. 1 Closed subbases It will be assumed that all spaces are 2\. Main topics are: compactness of convexity structures, continuity of convex closure operators, and characterization of convex sets. In the third part we characterize the Samuel realcompact spaces, that is, those uniform spaces which are complete with the weak uniformity induced by all the real-valued uniformly continuous functions on the space. SUBBASES, CONVEX SETS, AND HYPERSPACES JAN VAN MILL, MARCEL VAN DE VEL We introduce a notion of topological convexity.
Then (2Gn:n GN) X is sequenc a oef non-empt y open subset of 2X.As s 2 is i-pseudocompact, this sequenc has -limi ae t poin F Gt2 X.Choos ane y point p £ F. Thus, let (Gn: n GN) be a sequence of non-empty open subset osf X.
Hung, Characterization of bases of countable order and factorization of monotone developability, Top. Suppos 2X ies i-pseudocompact W.e show tha X it «-pseudos - compact. Next, we characterized those metric spaces which are metrizable by a Bourbaki-complete metric or cofinally Bourbaki-complete metric. Künzi, Properties related to compactness in hyperspaces, Top. In the second part, we embed the Bourbaki-complete metric spaces (and after, the Bourbaki-complete uniform spaces) into an universal space. Moreover, we show that completeness and cofinal completeness are, respectively, equivalent to Bourbaki-completeness and cofinal Bourbaki-completeness whenever we consider uniformities having a base of star-finite covers. In particular, we solve several primary problems related to products, subspaces, hyperspaces, metric spaces and fine spaces. In the first part of the thesis we study Bourbaki-completeness and cofinal Bourbaki-completeness (equivalently, uniform strong-paracompactness) of uniform spaces. It is divided in three parts where the related topics Bourbaki-completeness and Samuel realcompactification are studied. This thesis belongs to the area of General Topology and, in particular, to the field of study of uniform spaces. L K be a compact convex subset of a metriz- able locally convex real topological vector space L, dimK 2.
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