Condensations of Cartesian products

We consider when one-to-one continuous mappings can improve normalitytype and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space X there is a μ such that X can be condensed onto a normal (σ-compact) space if and only if there is no measurable cardinal. For any Tychonoff space X and any cardinal ν there is a Tychonoff space M which preserves many properties of X and such that any one-to-one continuous image of M, μ ≤ ν, contains a closed copy of X. For any infinite compact space K there is a normal space X such that X ×K cannot be mapped one-to-one onto a normal space.