zbMATH — the first resource for mathematics

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

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