A note on the Brouwer dimension of chainable spaces

A general method produces from a compact Hausdorff space S a compact Hausdorff space T with Ind T = Ind S + 1 . We show that if S is chainable, then T is also chainable while Dg T < Ind T , where Dg denotes dimensionsgrad, the dimension in the original sense of Brouwer. This leads to a chainable, first countable, separable space X n with Dg X n < Ind X n = n for each integer n > 1 .