On the shape category of compacta

Abstract

K. Borsuk [Fund. Math. 99 (1978), no. 1, 35–42] extended the classical notion of Lyusternik-Shnirelman category (briefly L-S category) to the theory of shape and thereby introduced a shape invariant coefficient for a compactum X . This was subsequently generalized for all topological spaces by V. V. Agaronyan and Yu. M. Smirnov [Soviet Math. Dokl. 20 (1979), no. 3, 516–518] and Z. Čerin [Houston J. Math. 5 (1979), no. 2, 169–182]. The author studies the shape L-S category of compacta and proves, among other things, that the coefficients of two compacta are equal provided there exists a refinable map between them. A new shape invariant called "the coefficient of movability'' is given and some of its properties for shape fibrations are studied. Some properties of Borsuk's coefficient λ(X) introduced in his aforementioned paper are given. Several relations, equalities and inequalities, between these coefficients are also given.