In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ), defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces which make them Hausdorff topological groups. We then exhibit an example in whichπ top k succeeds in distinguishing the shape type of X and Y whileπ k fails, for all k ∈ N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity ofπ top k with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the kth shape group induced by the kth shape loop space and show that it coincides with the above topology.
The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial bundles. In particular, we determine the structure of the n-dimensional Hawaiian group of the m-dimensional Hawaiian earring space, for all 1 ≤ m ≤ n.
K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and asked some questions concerning properties of the capacity of compacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, we obtain the capacity of a Moore space M(A, n) and an Eilenberg-MacLane space K(G, n). Also, we compute the capacity of the wedge sum of finitely many Moore spaces of different degrees and the capacity of the product of finitely many Eilenberg-MacLane spaces of different homotopy types. In particular, we give exact capacity of the wedge sum of finitely many spheres of the same or different dimensions.
D. K. Biss (Topology and its Applications 124 (2002) [355][356][357][358][359][360][361][362][363][364][365][366][367][368][369][370][371] introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.2000 Mathematics Subject Classification. 55Q05; 55U40; 54H11; 55P35.
K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and raised some interesting questions about it. In this paper, during computing the capacity of wedge sum of finitely many spheres of different dimensions and the complex projective plane, we give a negative answer to a question of Borsuk whether the capacity of a compactum determined by its homology properties.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.