Thirty years of shape theory

The paper outlines the development of shape theory since its founding by K. Borsuk 30 years ago to the present days. As a motivation for introducing shape theory, some shortcomings of homotopy theory in dealing with spaces of irregular local behavior are described. Special attention is given to the contributions to shape theory made by the Zagreb topology group

On iterated inverse limits

AbstractLet a compact Hausdorff space X be the limit of a cofinite inverse system of compact Hausdorff spaces Xλ, X=limλXλ. Then it is possible to express every Xλ as the limit of an inverse system of compact polyhedra Yλμ, Xλ=limμYλμ, in such a way that the spaces Yν=Yλμ can be organized in an inverse system with limνYν=limλ limμYλμ. Using ANR-resolutions, the result is generalized to non-compact spaces

The topological dimension of type I C*-algebras

While there is only one natural dimension concept for separable, metric spaces, the theory of dimension in noncommutative topology ramifies into different important concepts. To accommodate this, we introduce the abstract notion of a noncommutative dimension theory by proposing a natural set of axioms. These axioms are inspired by properties of commutative dimension theory, and they are for instance satisfied by the real and stable rank, the decomposition rank and the nuclear dimension. We add another theory to this list by showing that the topological dimension, as introduced by Brown and Pedersen, is a noncommutative dimension theory of type I C*-algebras. We also give estimates of the real and stable rank of a type I C*-algebra in terms of its topological dimension.Comment: 20 pages; minor correction

On the singular homology of one class of simply-connected cell-like spaces

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called {\sl Snake space}. In the sequel we introduced the functor $SC(-,-)$ defined on the category of all spaces with base points and continuous mappings. For the circle $S^1$, the space $SC(S^1, \ast)$ is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces $SC(Z, \ast)$ for any path-connected compact spaces $Z$

Thirty years of shape theory

The paper outlines the development of shape theory since its founding by K. Borsuk 30 years ago to the present days. As a motivation for introducing shape theory, some shortcomings of homotopy theory in dealing with spaces of irregular local behavior are described. Special attention is given to the contributions to shape theory made by the Zagreb topology group