Banach–Stone type theorems on uniformly continuous and Lipschitz continuous pseudometrics

Abstract For a uniform space $$Z = (Z,\mathcal {U}_Z)$$ Z = ( Z , U Z ) with a uniformity $$\mathcal {U}_Z$$ U Z , a pseudometric d on Z is uniformly continuous with respect to $$\mathcal {U}_Z$$ U Z if for each $$\epsilon > 0$$ ϵ > 0 , there is an entourage $$U \in \mathcal {U}_Z$$ U ∈ U Z such that if $$(x,y) \in U$$ ( x , y ) ∈ U , then $$d(x,y) < \epsilon $$ d ( x , y ) < ϵ . When $$Z = (Z,d_Z)$$ Z = ( Z , d Z ) is a metric space with an admissible metric $$d_Z$$ d Z , d is Lipschitz continuous with respect to $$d_Z$$ d Z if there exists $$k > 0$$ k > 0 such that for every pair $$(x,y) \in Z^2$$ ( x , y ) ∈ Z 2 with $$x \ne y$$ x ≠ y , $$d(x,y) \le kd_Z(x,y)$$