Some generalizations of rapid ultrafilters in topology and Id-fan tightness

Abstract

In this paper, we introduce the weakly $k$-rapid points,for $1\leqq k \lt \omega$, and the rapid points of topological spaces. They extend the concept of rapid ultrafilter. It is evident from the definition that every weak $P$-point is a rapid point and a weakly $k$-rapid point for $1\leqq k \lt \omega$. We show: (a) there is a space containing a rapid, non-weak-$P$-point $\Leftrightarrow$ there is a rapid ultrafilter on $\omega$; and (b) there is a space containing a weakly $k$-rapid, non-weak-$P$-point, for some $ 1\leqq k \lt \omega\Leftrightarrow$ there is a $Q$-point in $\beta(\omega)\backslash \omega\Leftrightarrow$ for every $ 1\leqq k \lt \omega$,there is a space which is weakly $(k+1)$ -rapid and is not weakly $k$-rapid. Assuming the existence of a $Q$-point in $\beta(\omega)\backslash \omega$,we give an example of a zero-dimensional homogeneous space without weak $P$-points such that all its points are rapid. Finally, the concept of Id-fan tightness is introduced as a generalization of countable strong fan tightness.