On uniform connectedness
Summary: The concept of uniform connectedness, which generalizes the concept of well-chainedness for metric spaces, is used to prove the following: (a) If two points a and b of a compact Hausdorff uniform space (X,\({\mathcal U})\) can be joined by a U-chain for every \(U\in {\mathcal U}\), then they lie together in the same component of X; (b) Let (X,\({\mathcal U})\) be a compact Hausdorff uniform space, A and B non-empty disjoint closed subsets of X such that no component of X intersects both A and B. Then there exists a separation \(X=X_ A\cup X_ B\), where \(X_ A\) and \(X_ B\) are disjoint compact sets containing A and B respectively. These generalize the corresponding results for metric spaces.
scientific article; zbMATH DE number 4103954