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3 Bulletin of the Polish Academy of Sciences. Mathematics

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Spaces of upper semi-continuous multi-valued functions which are absolute retracts

Uehara S.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 3

319-328

Let X = (X, d] be a metric space. We endow the hyperspace S^X x R consisting of non-empty closed subsets of X x R with the topology induced by d_H defined by d_H(E,F) = inf{epsilon is an element of (0,infinity] | N(E,epsilon) is a subset of F and N(F,epsilon) is a subset E}. Let USCC(X) be a space of upper semi-continuous multi-valued functions phi : X --> R such that phi (x) is a closed interval for every x is an element of X. Identifying those functions with their graphs, we consider USCC(X) as a subspace of 2^X x R. We give a necessary and sufficient condition on X is order that USCC(X) is closed in 2^X x R. In case X is complete, we also give a necessary and sufficient condition on USCC_B(X) to be an AR, where USCC_B(X) is a subspace of USCC(X) consisting of all bounded functions. As a corollary, we find that USCC(X) is an AR if X is compact.

On local n-connectedness in hyperspaces

Makuchowski W.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

29-33

It is shown that for any compact metric space X in hyperspaces C(X) of subcontinua of X and 2[sup X] of all nonempty closed subsets of X, local connectedness and local n-connectedness are equivalent at any point.

On local connectedness in hyperspaces

Makuchowski W.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 2

119-126

It is shown that for any Hausdorff compactum X in the hyperspace C(X) of subcontinua of X (in the hyperspace 2^X of all nonempty closed subsets of X) local connectedness and local arcwise connectedness are equivalent at any point. If X has the property of Kelley, then local connectedness of C(X) is proved to be equivalent to some stronger kinds of local connectedness and of local arcwise connectedness.