## Tsukuba Journal of Mathematics

### Countable product of function spaces having p-Frechet-Urysohn like properties

Angel Tamariz-Mascarua

#### Abstract

We exhibit in this article some classes of spaces for which properties $\gamma$ and $\gamma_{p}$ are countable additive, and we prove that for some type of spaces and ultrafilters $p\in\omega^{*}, \gamma$ is equivalent to $\gamma_{p}$. We obtain: (1) If $\{X_{n}\}_{n \lt \omega}$ is a sequence of metrizable locally compact spaces with $\gamma_{p}(p\in\omega^{*})$, then $\Pi_{n \lt \omega}C_{\pi}(X_{n})$ is a $FU(p)$-space; (2) $C_{\pi}(X)$ is a Fréchet-Urysohn (resp., $FU(p)$) space iff $C_{\pi}(F(X))$ has the same property, where $F(X)$ is the free topological group generated by $X$; (3) For a locally compact metrizable and non countable space $X$, $C_{\pi}(X)$ is a Fréchet-Urysohn (resp., $FU(p)$) space iff $C_{\pi}(L_{\pi}(X))$ is Fréchet-Urysohn (resp., $FU(p)$), where $L_{\pi}(X)$ is the dual space of $C_{\pi}(X)$; (4) For every $\check{C}$ech complete space $X$ and every $p\in\omega^{*}$ for which $R$ does not have $\gamma_{p}$, $C_{\pi}(X)$ is Fréchet-Urysohn iff $C_{\pi}(X)$ is a $FU(p)$-space. Also we give some results concerning $P$-points in $\omega^{*}$ related with $p$-Fréchet-Urysohn property and topological function spaces.

#### Article information

Source
Tsukuba J. Math., Volume 20, Number 2 (1996), 291-319.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496163083

Digital Object Identifier
doi:10.21099/tkbjm/1496163083

Mathematical Reviews number (MathSciNet)
MR1422622

Zentralblatt MATH identifier
0888.54019

#### Citation

Tamariz-Mascarua, Angel. Countable product of function spaces having p-Frechet-Urysohn like properties. Tsukuba J. Math. 20 (1996), no. 2, 291--319. doi:10.21099/tkbjm/1496163083. https://projecteuclid.org/euclid.tkbjm/1496163083