On compact weaker topologies in function spaces

Abstract

In this paper we prove that for every cardinal kappa, the space C-p(D-k) admits a continuous bijection onto a space whose all finite powers are Lindelof (the symbol D stands for the discrete two-point space). We also prove that for every metrizable compact space X, the space Cp (X) can be condensed (i.e., admits a continuous bijection) onto the Hilbert cube I-omega. As a consequence it is established that the space Cp (DI) can be condensed onto a compact space. In connection to this result, we also prove that there exist models of ZFC in which the statement "The spaces C-p(D-k) can be condensed onto a compact space for every cardinal kappa > omega" is not true. We show also that for every cardinal K, the spaces C-p(C-p(D-k)) and L-p(D-k) have dense subsets of countable tightness. (C) 2001 Elsevier Science B.V. All rights reserved.