Consonance and Cantor set-selectors
It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.
Trivial bundles of spaces of probability measures and countable-dimensionality
The probability measure functor P carries open continuous mappings of compact metric spaces into Q-bundles provided Y is countable-dimensional and all fibers are infinite. This answers a question raised by V. Fedorchuk.
Strongly paracompact metrizable spaces
Strongly paracompact metrizable spaces are characterized in terms of special S-maps onto metrizable non-Archimedean spaces. A similar characterization of strongly metrizable spaces is obtained as well. The approach is based on a sieve-construction of "metric"-continuous pseudo-sections of lower semicontinuous mappings.
Weak selections and weak orderability of function spaces
It is proved that for a zero-dimensional space , the function space has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space , the function space is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial...
Weak orderability of second countable spaces
We demonstrate that a second countable space is weakly orderable if and only if it has a continuous weak selection. This provides a partial positive answer to a question of van Mill and Wattel.
Closed graph multi-selections
A classical Lefschetz result about point-finite open covers of normal spaces is generalised by showing that every lower semi-continuous mapping from a normal space into the nonempty compact subsets of a metrizable space admits a closed graph multi-selection. Several applications are given as well.
Selections and approaching points in products
The present paper aims to furnish simple proofs of some recent results about selections on product spaces obtained by García-Ferreira, Miyazaki and Nogura. The topic is discussed in the framework of a result of Katětov about complete normality of products. Also, some applications for products with a countably compact factor are demonstrated as well.
Hyperspace selections avoiding points
We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...
Classical-type characterizations of non-metrizable ANE(n)-spaces
The Kuratowski-Dugundji theorem that a metrizable space is an absolute (neighborhood) extensor in dimension n iff it is (resp., ) is extended to a class of non-metrizable absolute (neighborhood) extensors in dimension n. On this base, several facts concerning metrizable extensors are established for non-metrizable ones.
Does C* -embedding imply C*-embedding in the realm of products with a non-discrete metric factor?
The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.
Selections generating new topologies.
Every (continuous) selection for the non-empty 2-point subsets of a space X naturally defines an interval-like topology on X. In the present paper, we demonstrate that, for a second-countable zero-dimensional space X, this topology may fail to be first-countable at some (or, even any) point of X. This settles some problems stated in [7].
Open maps having the Bula property
An open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F₀,F₁ ⊂ X with f(F₀) = Y = f(F₁), provided all fibers of f are infinite and C*-embedded in X. Applications are given to the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.
Weak selections and flows in networks
We demonstrate that every Vietoris continuous selection for the hyperspace of at most 3-point subsets implies the existence of a continuous selection for the hyperspace of at most 4-point subsets. However, in general, we do not know if such ``extensions'' are possible for hyperspaces of sets of other cardinalities. In particular, we do not know if the hyperspace of at most 3-point subsets has a continuous selection provided the hyperspace of at most 2-point subsets has a continuous selection.
Continuous selections, -subsets of Banach spaces and usco mappings
Every l.s.cṁapping from a paracompact space into the non-empty, closed, convex subsets of a (not necessarily convex) -subset of a Banach space admits a single-valued continuous selection provided every such mapping admits a convex-valued usco selection. This leads us to some new partial solutions of a problem raised by E. Michael.
Factorizations of set-valued mappings with separable range
Right factorizations for a class of l.s.cṁappings with separable metrizable range are constructed. Besides in the selection and dimension theories, these l.s.cḟactorizations are also successful in solving the problem of factorizing a class of u.s.cṁappings.
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