On spaces with connected Higson coronas (Q2630461)

It is well known that if \((X,d)\) is a metric space, its metric topology can also be generated by a metric with arbitrarily small distances, \(d_\epsilon(x,y) = \min(d(x,y),\epsilon)\). This fact can be interpreted as saying that (metric) topology concerns itself only with the small-scale structure of a metric space. There is a complementary concept of \textit{coarse structure}: intuitively, this is the part of the structure of a metric space which is invariant under changes of small size. For instance, coarse structure does not distinguish between \(\mathbb Z\) and \(\mathbb R\), which in this sense do not differ at a scale below one unit; but it distinguishes between the (topologically equivalent) \(\mathbb Z\) and \(\mathbb N\), because one of them approaches infinity in two directions, the other only in one. The \textit{Higson compactification} of a metric space \(X\) is a compact extension of \(X\) that shares the coarse structure of the original space; the part that is added is called the \textit{Higson corona}. This paper shows that the Higson corona is connected if every coarse map from \(X\) to a coarse coproduct \(Y + Z\) is close to a coarse map into one summand. Various related results are also proved.