NUMBER AND PERFECT MAPS

The metrizability number, m(X), of a space X is the least cardinality of a cover of X by metrizable subspaces. We show that if f : X → Y is a perfect map, and m(X) < ω, then m(Y ) ≤ ω, and we give an example where 2 = m(X) < m(Y ) = ω. The example shows that, in the finite case, the metrizability number can be increased by a perfect mapping; this was shown by Ismail and Szymanski not to be the case in the realm of locally compact spaces. It remains an open question whether or not an infinite metrizability number can be increased by a perfect mapping; in particular, the case m(X) = ω is unsolved.