## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 19, Number 1 (1995), 173-185.

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

Salvador Garcia-Ferreira and Angel Tamariz-Mascarua

#### 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.

#### Article information

**Source**

Tsukuba J. Math., Volume 19, Number 1 (1995), 173-185.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496162806

**Digital Object Identifier**

doi:10.21099/tkbjm/1496162806

**Mathematical Reviews number (MathSciNet)**

MR1346760

**Zentralblatt MATH identifier**

0835.54023

**Subjects**

Primary: 54A35: Consistency and independence results [See also 03E35]

Secondary: 54D80: Special constructions of spaces (spaces of ultrafilters, etc.) 54G15: Pathological spaces

**Keywords**

rapid ultrafilter $Q$-point semiselective rapid point weakly $k$-rapid point countable strong fan tightness Id-fan tightness countable fan tightness

#### Citation

Garcia-Ferreira, Salvador; Tamariz-Mascarua, Angel. Some generalizations of rapid ultrafilters in topology and Id-fan tightness. Tsukuba J. Math. 19 (1995), no. 1, 173--185. doi:10.21099/tkbjm/1496162806. https://projecteuclid.org/euclid.tkbjm/1496162806