On quasidomination of compacta

Abstract

The notions of quasidomination and quasi-equivalence of compacta were introduced by K. Borsuk [Fund. Math. 93 (1976), no. 3, 197–212. These relations are weaker than the relations of shape domination and shape equivalence. According to Borsuk they allow the consideration of shape from a quantitative point of view. In this note the author studies several properties of quasidomination in connection with some shape invariants and X -likeness. For example, he obtains the following generalization of a result of Borsuk. Theorem: Let X and Y be two compacta lying in the Hilbert cube. If Y is movable and X -like, then Y is quasidominated by X . Furthermore, he obtains a simple characterization of quasidomination of movable compacta. He also shows that if an FANR compactum Y is quasidominated by X , then Y is shape dominated by X . The author points out that this last result can also be obtained by using the work of L. Boxer and R. B. Sher [Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 26 (1978), no. 9-10, 849–853.