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On uniform connection properties. (English) Zbl 0558.54018
A uniform space (X,\({\mathcal U})\) has property S (is said to be uniformly locally connected) iff for any entourage \(U\in {\mathcal U}\) there exists a finite covering of X formed by connected U-small subsets of X (for each \(U\in {\mathcal U}\) there exists \(V\in {\mathcal U}\) such that \(V\in U\) and V(x) is connected for each \(x\in X)\). The authors prove that the uniform local connectedness is preserved by uniform quotient maps, a uniform product has the property S (is uniformly locally connected) iff each factor has the corresponding property and all but finitely many factors are connected, and also, that a uniform space has the property S iff its coreflection within the subcategory of all uniformly locally connected spaces) has the property S.
Reviewer: J.Chvalina

54E15 Uniform structures and generalizations
54D05 Connected and locally connected spaces (general aspects)
54B30 Categorical methods in general topology
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