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 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 C(f), Q(f), E(f) and A(f) be the sets of all continuity, quasicontinuity, upper and lower quasicontinuity and cliquishness points of a real function f: X → ℝ, respectively. The triplets (C(f),Q(f),A(f)), (C(f),E(f),A(f) and (Q(f),E(f),A(f)are characterized for functions defined on Baire metric spaces without isolated points.

It is known that the ring $B\left(ℝ\right)$ of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring $C\left(ℝ\right)$ of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of $C\left(ℝ\right)$. In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of $C\left(ℝ\right)$ which differs from $B\left(ℝ\right)$.

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on ${\left[\omega \right]}^{\omega }$, ${\left(s\right)}_0$ is a proper subset of the hereditary ideal associated with (s).  We construct an example in the Ellentuck topology of a set which is...

We study the possibility of extending any bounded Baire-one function on the set of extreme points of a compact convex set to an affine Baire-one function and related questions. We give complete solutions to these questions within a class of Choquet simplices introduced by P. J. Stacey (1979). In particular we get an example of a Choquet simplex such that its set of extreme points is not Borel but any bounded Baire-one function on the set of extreme points can be extended to an affine Baire-one function....

Let $A\left(X\right)$ denote a subalgebra of $C\left(X\right)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A\left(X\right)$, we generalize the notion of $z_A^{\beta }$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C\left(X\right)$ containing...

In connection with a conjecture of Scheepers, Bukovský introduced properties wQN* and SSP* and asked whether wQN* implies SSP*. We prove it in this paper. We also give characterizations of properties S₁(Γ,Ω) and $S_{fin}(\Gamma ,\Omega )$ in terms of upper semicontinuous functions

Suppose that $(X,𝒜)$ is a measurable space and $Y$ is a metrizable, Souslin space. Let ${𝒜}^u$ denote the universal completion of $𝒜$. For $x\in X$, let $\underline{f}(x,·)$ be the lower semicontinuous hull of $f(x,·)$. If $f:X\times Y\to \overline{ℝ}$ is $({𝒜}^u\otimes ℬ\left(Y\right),ℬ\left(\overline{ℝ}\right))$-measurable, then $\underline{f}$ is $({𝒜}^u\otimes ℬ\left(Y\right),ℬ\left(\overline{ℝ}\right))$-measurable.