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

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.