Some results and problems about weakly pseudocompact spaces

summary:A space $X$ is {\it truly weakly pseudocompact} if $X$ is either weakly pseudocompact or Lindelöf locally compact. We prove: (1) every locally weakly pseudocompact space is truly weakly pseudocompact if it is either a generalized linearly ordered space, or a proto-metrizable zero-dimensional space with $\chi (x,X)>\omega$ for every $x\in X$; (2) every locally bounded space is truly weakly pseudocompact; (3) for $\omega < \kappa <\alpha$, the $\kappa$-Lindelöfication of a discrete space of cardinality $\alpha$ is weakly pseudocompact if $\kappa = \kappa ^\omega$

Disconnectedness properties of hyperspaces

summary:Let $X$ be a Hausdorff space and let $\mathcal H$ be one of the hyperspaces $CL(X)$, $\mathcal K(X)$, $\mathcal F(X)$ or $\mathcal F_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathcal H$: extremal disconnectedness, being a $F'$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $\mathcal K(X)$ is hereditarily disconnected. We also show an example proving that this result cannot be reversed

pp-sequential like properties in function spaces

summary:We introduce the properties of a space to be strictly $\operatorname{WFU}(M)$ or strictly $\operatorname{SFU}(M)$, where $\emptyset \neq M\subset \omega ^{\ast }$, and we analyze them and other generalizations of $p$-sequentiality ($p\in \omega ^{\ast }$) in Function Spaces, such as Kombarov's weakly and strongly $M$-sequentiality, and Kocinac's $\operatorname{WFU}(M)$ and $\operatorname{SFU}(M)$-properties. We characterize these in $C_\pi (X)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $\operatorname{WFU}(L(M))$-property, where $L(M)=\{{}^{\lambda }p:\lambda <\omega _1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset \omega ^{\ast }$; and that $C_\pi (X)$ is a $\operatorname{WFU}(L(M))$-space iff $X$ satisfies the $M$-version $\delta _M$ of Gerlitz and Nagy's property $\delta$. We also prove that if $C_\pi (X)$ is a strictly $\operatorname{WFU}(M)$-space (resp., $\operatorname{WFU}(M)$-space and every $\operatorname{RK}$-predecessor of $p\in M$ is rapid), then $X$ satisfies $C''$ (resp., $X$ is zero-dimensional), and, if in addition, $X\subset \Bbb R$, then $X$ has strong measure zero (resp., $X$ has measure zero), and we conclude that $C_\pi (\Bbb R)$ is not $p$-sequential if $p\in \omega ^{\ast }$ is selective. Furthermore, we show: (a) if $p\in \omega ^{\ast }$ is selective, then $C_\pi (X)$ is an $\operatorname{FU}(p)$-space iff $C_\pi (X)$ is a strictly $\operatorname{WFU}(T(p))$-space, where $T(p)$ is the set of $\operatorname{RK}$-equivalent ultrafilters of $p$; and (b) $p\in \omega ^{\ast }$ is semiselective iff the subspace $\omega \cup \{p\}$ of $\beta \omega$ is a strictly $\operatorname{WFU}(T(P))$-space. Finally, we study these properties in $C_\pi (Z)$ when $Z$ is a topological product of spaces