Characterizing cohomological dimension: The cohomological dimension of \(A \cup{} B\) (Q1187112)

The Menger-Urysohn sum formula is proved for cohomological dimension with respect to integers. The proof is based on a characterization of cohomological dimension \(\dim_{\mathbb{Z}}\) given by the author for metrizable spaces. That characterization is the natural transfer of Edward's characterization of \(\dim_{\mathbb{Z}}\) for compact metric spaces. It turns out that the proof of the Menger-Urysohn formula with this approach is rather long and complicated. A shorter proof (of a more general theorem) can be found in an outgoing paper by \textit{J. Dydak} [Cohomological dimension of metrizable spaces II, Trans. Am. Math. Soc. (to appear)].