The Smirnov remainders of uniformly locally connected proper metric spaces

The aim of this paper is to investigate relations between uniform local connectedness and the dimension of the Smirnov remainder. In particular, we devote this paper to calculating the dimension of the Smirnov remainder u d R n ∖ R n of the n-dimensional Euclidean space ( R n , d ) with uniform local connectedness. We show that dim u d R ∖ R = ind u d R ∖ R = Ind u d R ∖ R = 1 if ( R , d ) is uniformly locally connected. Moreover, we introduce a new concept of “thin” covering spaces, and we have the following: If an infinite covering space ( R 2 , d ˜ ) of a compact 2-manifold is “thin”, then dim u d ˜ R 2 ∖ R 2 = ind u d ˜ R 2 ∖ R 2 = Ind u d ˜ R 2 ∖ R 2 = 2 .