38 research outputs found
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 defined on the category of all
spaces with base points and continuous mappings. For the circle , the
space is a Snake space. In the present paper we study the
higher-dimensional homology and homotopy properties of the spaces
for any path-connected compact spaces