Self-maps under the compact-open topology

Abstract

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