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 = indu(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) over tilde) of a compact 2-manifold is thin then dim u((d) over tilde)R(2) \ R(2) = indu((d) over tilde)R(2) \ R(2) = Ind u((d) over tilde) \ R(2) = 2 (C) 2010 Elsevier B V All rights reserved