The point of continuity property, neighbourhood assignments and filter convergences (Q2915370)

A function \(f:X\to Z\) has the \textit{point of continuity property} (PCP) if the restriction of \(f\) to each non-empty closed subset of \(X\) has a point of continuity. If \((Z, d)\) is a metric space, then we say that \(f:X\to Z\) is \textit{weakly separated} if for each \(\varepsilon>0\) there is a neighbourhood assignment \((V_x)_{x\in X}\) of \(X\) such that \(d(f(x),f(y))\leq\varepsilon\) whenever \((x,y)\in V_y\times V_x\). It is known that for a real-valued function \(f\) defined on a Polish space \(X\) the following conditions are equivalent: (1)~\(f\) is weakly separated; (2)~\(f\) has PCP; (3)~\(f\) is \(F_\sigma\)-measurable; (4)~\(f\) is of the class \(B_1\), see \textit{P.-Y.~Lee, W.-K.~Tang} and \textit{D.~Zhao} [Proc. Am. Math. Soc. 129, No. 8, 2273--2275 (2001; Zbl 0970.26004)].NEWLINENEWLINEThe paper under review can be divided into two parts. In the first part the author shows that the implication \((1)\Rightarrow (2)\) holds also for \(f\) defined on any hereditarily Baire space \(X\) satisfying one of the following conditions: (i)~\(X\) is monotonically semistratifiable; (ii)~\(X\) is a suborderable monotonic \(\beta\)-space; (iii)~\(X\) is a monotonic \(\beta\)-space and has a point-countable \(T_0\)-separating open collection. It follows from (i) that every weakly separated function defined on a hereditarily Baire semistratifiable space is \(\sigma\)-discrete in Hansell's sense and \(F_\sigma\)-measurable. The results also imply that every separately continuous function \(f:X\times Y\to Z\) defined on a hereditarily Baire product of two semistratifiable spaces \(X\) and \(Y\) into a metric space \(Z\) is \(F_\sigma\)-measurable and \(\sigma\)-discrete. This answers in the affirmative a question posed by T.~Banakh and V.~K.~Maslyuchenko; see [\textit{T. O. Banakh}, Mat. Stud. 18, No. 1, 10--28 (2002; Zbl 1023.54023)].NEWLINENEWLINEIn the second part the author works with the question: for which filters \(\mathcal{F}\) on the integers is the \(\mathcal{F}\)-limit of any \(\mathcal{F}\)-convergent sequence of continuous functions weakly separated? He proves that for every filter \(\mathcal{F}\) the following are equivalent: (a)~\(\mathcal{F}\) is \(\omega\)-diagonalizable by \(\mathcal{F}\)-universal sets (in the sense of Laflamme); (b)~\(\mathcal{F}\) is \(F_\sigma\)-separated from its dual ideal; (c)~the \(\mathcal{F}\)-limit of any sequence of continuous functions is \(F_\sigma\)-measurable; (d)~the \(\mathcal{F}\)-limit of any sequence of continuous functions is weakly separated. Similar results on analytic filters were obtained recently by \textit{M.~Laczkovich} and \textit{I.~Recław} [Fundam. Math. 203, No. 1, 39--46 (2009; Zbl 1172.03025)] and by \textit{G.~Debs} and \textit{J.~Saint Raymond} [ibid. 204, No. 3, 189--213 (2009; Zbl 1179.03046)].