Cell-like maps onto non-compact spaces of finite cohomological dimension (Q1103909)

It is shown that a metrizable space X is the cell-like image of a metrizable space Z if and only if Z has finite integral cohomological dimension. This extends the earlier result of J. J. Walsh and R. D. Edwards on the equivalence of the cell-like dimension raising map question for compact metric spaces with the question of whether or not all compact metric spaces having finite integral cohomological dimension must have finite covering dimension to general metric spaces. In a brilliant piece of work, A. N. Dranishnikov has recently shown the existence of an infinite-dimensional compact metric space whose integral cohomological dimension is equal to three.