Discrete $n$-tuples in Hausdorff spaces
Volume 187 / 2005
Abstract
We investigate the following three questions: Let $n\in {\mathbb N}$. For which Hausdorff spaces $X$ is it true that whenever ${\mit \Gamma }$ is an arbitrary (respectively finite-to-one, respectively injective) function from ${\mathbb N}^n$ to $X$, there must exist an infinite subset $M$ of ${\mathbb N}$ such that ${\mit \Gamma }[M^n]$ is discrete? Of course, if $n=1$ the answer to all three questions is “all of them”. For $n\geq 2$ the answers to the second and third questions are the same; in the case $n=2$ that answer is “those for which there are only finitely many points which are the limit of injective sequences”. The answers to the remaining instances involve the notion of n-Ramsey limit. We also show that the class of spaces satisfying these discreteness conclusions for all $n$ includes the class of F-spaces. In particular, it includes the Stone–Čech compactification of any discrete space.
