1,720,999 research outputs found
Consonance and Cantor set-selectors
Get PDFIt 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.peer-reviewe
Insertion of Continuous Set-Valued Mappings
Get PDFAn interesting result about the existence of "intermediate" set-valued
mappings between pairs of such mappings was obtained by Nepomnyashchii. His
construction was for a paracompact domain, and he remarked that his result is
similar to Dowker's insertion theorem and may represent a generalisation of
this theorem. In the present paper, we characterise the -paracompact
normal spaces by this set-valued "insertion" property and for ,
i.e. for countably paracompact normal spaces, we show that it is indeed
equivalent to the mentioned Dowker's theorem. Moreover, we obtain a similar
result for -collectionwise normal spaces and show that for normal spaces,
i.e. for -collectionwise normal spaces, our result is equivalent to the
Kat\v{e}tov-Tong insertion theorem. Several related results are obtained as
well
Generic extensions of finite-valued u.s.c. selections
No full textAbstractAs a rule, most of the classical Michael-type selection theorems are analogues and, in some respects, generalizations of ordinary extension theorems. In this paper we show that the existence of set-valued u.s.c. selections for l.s.c. mappings is not related to the “usual” mapping-extension problem for u.s.c. mappings. In view of that, the paper is especially devoted to a proper notion of extending u.s.c. mappings that agrees well with the existing selection results. On this base new selection theorems dealing with controlled u.s.c. “extensions” of partial u.s.c. selections are obtained. Possible applications are illustrated in the dimension theory of normal spaces
Paracompactness and Open Relations
Get PDFThe countably paracompact normal spaces were characterised by Dowker and
Kat\v{e}tov in terms of an insertion property. Dowker also characterised them
by normality of their product with the closed unit interval. Michael used the
Dowker-Kat\v{e}tov insertion property to motivate his selection
characterisation of these spaces. Morita extended in a natural way Dowker's
product characterisation to all -paracompact normal spaces. In this
paper, we look at these results from the point of view of open relations.
Insertions and selections are equivalent for such relations. Furthermore, we
obtain a natural characterisation of -paracompact normal spaces in terms
of selections for convex-valued open relations. Based on this characterisation,
we give simple alternative proofs of the above mentioned results. Other
applications are obtained as well
Selections and approximations in finite-dimensional spaces
No full textAbstractThe paper presents a general approach to some selection results for set-valued mappings defined on finite-dimensional spaces. The approach is essentially based on open-graph mappings and demonstrates the topological genesis of approximate selections in finite-dimensional spaces. It culminates in a number of new applications taking in account the role of different hypotheses that are natural for such selection theorems
Selections and hyperspaces of finite sets
No full textAbstractIt is demonstrated that the hyperspace of at most (n+1)-point sets has a Vietoris continuous selection if both the hyperspace of at most n-point sets and that of exactly (n+1)-point sets have Vietoris continuous selections. This result is applied to demonstrate that the hyperspace of at most (2n+2)-point sets has a Vietoris continuous selection provided that one of at most (2n+1)-point sets has such a selection. This settles some open questions
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