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

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In this paper, we introduce the weakly k-rapid points, for 1[?]k<ω, 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[?]k<ω. We show: (a) there is a space cintaining a rapid, non-weak-P-point ⇔ there is a rapid ultrafilter on ω; and (b) there is a space containing a weakly k-rapid, non-weak-P-point, for some 1[?]k<ω ⇔ there is a Q-point in β(ω)/ω、⇔ for every 1[?]k<ω, there is a space with is weakly (k+1) rapid and is not weakly k-rapid. Assuming the existence of a Q-point in β(ω)/ω, we give an example of a zero-dimensional homogeneous space without weak P-points such that all its points aer rapid. Fainally, the concept of Id-fan tightness is introduced as a generalization of countable strong fan tightness.
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