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.

Recently, S. Mardešič, L. R. Rubin and T. Watanabe have developed a theory of approximate inverse systems and approximate resolutions, providing thus a new tool to study topological spaces. M. G. Charalambous then introduced a somewhat simpler but more general notion of approximate system. Subsequently, S. Mardešič showed, by a rather general and complicated construction, that the two notions of approximate systems (approximate resolutions) share all relevant properties of their limits (resolutions). This paper presents a new and rather simple construction with the same properties. Moreover, in the case of topologically complete approximate resolutions, uniqueness up to isomorphisms is established. At the end, it is indicated how one can extend this construction onto approximate mappings.