Self-maps under the compact-open topology

Abstract

This paper investigates the space $C_k({\omega }^{\ast },{\omega }^{\ast })$, the space of continuous self-maps on the Stone-Čech remainder of the integers, ${\omega }^{\ast }$, equipped with the compact-open topology. Our main results are that (1) $C_k({\omega }^{\ast },{\omega }^{\ast })$ is Baire, (2) Stone-Čech extensions of injective maps on $\omega$ form a dense set of weak $P$-points in $C_k({\omega }^{\ast },{\omega }^{\ast })$, (3) it is independent of ZFC whether $C_k({\omega }^{\ast },{\omega }^{\ast })$ contains $P$-points, and that (4) $C_k({\omega }^{\ast },{\omega }^{\ast })$ is not an $F$-space, but contains, as ${\omega }^{\ast }$, no non-trivial convergent sequences.

22 pages