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

• Autorzy: Uehara S.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

complete metric spaces; upper semi-continuous multi-valued function; absolute retract; hyperspace of closed subsets; Hausdorff metric

PL

przestrzeń metryczna zupełna; retrakt absolutny; hiperprzestrzeń podzbiorów domkniętych; metryka Hausdorffa

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 3
• Strony: 319--328
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Uehara S.

• Takamatsu National College of Technology, 355 Chokushi-Cho, Takamatsu City, Kagawa 761-8058 Japan

Bibliografia

• [1] O. Hanner, Some theorem on absolute neighbourhood retract, Ark. Mat., 1 (1951) 389-408.
• [2] K. Sakai, S. Uehara, A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl., 92 (1999) 107-118.
• [3] K. Sakai, S. Uehara, Spaces of upper semi-continuous multi-valued functions on complete metric spaces, Fund. Math., 160 (1999) 199-218.
• [4] H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math., 111 (1981) 247-262.