A.V. Arhangel'skii, O.I. Pavlov
A note on condensations of $C_p(X)$ onto compacta

Comment.Math.Univ.Carolinae 43,3 (2002) 485-492.

Abstract:A condensation is a one-to-one continuous mapping onto. It is shown that the space $C_p(X)$ of real-valued continuous functions on $X$ in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum $X$ (Theorem 19). However, there exists a non-metrizable compactum $X$ such that $C_p(X)$ condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.

Keywords: condensation, compactum, network, Lindel\"of space, topology of pointwise convergence, $\sigma $-compact space, Eberlein compactum, Corson compactum, Borel set, monolithic space, tightness
AMS Subject Classification: Primary 54A25, 54C35, 54A35