| Popis: |
The paper is a contribution to the study of separation, connectedness and compactness in the setting of Cauchy spaces. This study is based on the concept of closed subsets and the related closure operator as introduced by M. Baran in the setting of topological constructs, see e.g "The notion of closedness in topological categories, Comment. Math. Univ. Carolinae, 34, 1993, 383-397" or "Compactness , perfectness, separation, minimality and closedness with respect to closure operators, Applied Categorical Structiures, 10, 2002, 403-415". In this paper the application of these notions to the topological construct of Cauchy spaces and Cauchy continuous maps has been worked out and the relation to existing concepts on separation and compactness for Cauchy spaces is established. For connectedness the new notion (expressing the fact that there are no nontrivial clopen subsets) coincides with the property that any morphism to a discrete object is constant. Compactness through the related closure operator is weaker than saying that the underlying convergence structure is compact. For T0 separation several variants are proposed, one of which coincides with saying that there is no indiscrete subspace with at least two points. |
