We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff $X$ satisfies...

We prove that if the topology on the set Seq of all finite sequences of natural numbers is determined by $P_{\lambda }$-filters and λ ≤ , then Seq is a $P_{\lambda }$-set in its Čech-Stone compactification. This improves some results of Simon and of Juhász and Szymański. As a corollary we obtain a generalization of a result of Burke concerning skeletal maps and we partially answer a question of his.

We introduce and study the notion of pairwise monotonically normal space as a bitopological extension of the monotonically normal spaces of Heath, Lutzer and Zenor. In particular, we characterize those spaces by using a mixed condition of insertion and extension of real-valued functions. This result generalizes, at the same time improves, a well-known theorem of Heath, Lutzer and Zenor. We also obtain some solutions to the quasi-metrization problem in terms of the pairwise monotone normality.

We consider a multifunction $F:T\times X\to 2^E$, where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

We introduce partial dcpo’s and show their some applications. A partial dcpo is a poset associated with a designated collection of directed subsets. We prove that (i) the dcpo-completion of every partial dcpo exists; (ii) for certain spaces $X$, the corresponding partial dcpo’s of continuous real valued functions on $X$ are continuous partial dcpos; (iii) if a space $X$ is Hausdorff compact, the lattice of all S-lower semicontinuous functions on $X$ is the dcpo-completion of that of continuous real valued...

Let ${\left\{X_{\alpha }\right\}}_{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda$. Suppose $X$ is the quotient space of the disjoint union of $X_{\alpha }$’s by identifying $x_{\alpha }$’s as one point $\sigma$. We try to characterize ideals of $C\left(X\right)$ according to the same ideals of $C\left(X_{\alpha }\right)$’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let $m$ be an infinite cardinal. (1) Is there any ring $R$ and $I$ an ideal in $R$ such that $I$ is an irreducible intersection of $m$ prime ideals? (2) Is there...

We prove that the maximal Hausdorff compactification $\chi f$ of a $T_2$-compactifiable mapping $f$ and the maximal Tychonoff compactification $\beta f$ of a Tychonoff mapping $f$ (see [P]) are perfect. This allows us to give a characterization of all perfect Hausdorff (respectively, all perfect Tychonoff) compactifications of a $T_2$-compactifiable (respectively, of a Tychonoff) mapping, which is a generalization of two results of Skljarenko [S] for the Hausdorff compactifications of Tychonoff spaces.

We show that a Tychonoff space is the perfect pre-image of a cofinally complete metric space if and only if it is paracompact and cofinally Čech complete. Further properties of these spaces are discussed. In particular, cofinal Čech completeness is preserved both by perfect mappings and by continuous open mappings.