Maslyuchenko, Oleksandr V. The discontinuity point sets of quasi-continuous functions. (English) Zbl 1123.54005 Bull. Aust. Math. Soc. 75, No. 3, 373-379 (2007). We call a subset \(E\) of a topological space \(X\) a junction set if there are subsets \(A\) and \(B\) of X such that \(\overline {A}\cap B = A\cap \overline{B} = \emptyset\) and \(E = \overline{A}\cap \overline{B}\). A function \(f:X\rightarrow Y\) is called quasi-continuous if \(f^{-1}(G) \subseteq \overline {\text{int}f^{-1}(G)}\) for each open \(G\subseteq Y\). We call a subset \(E\) of a topological space \(X\) a \(\sigma\)-junction set if there is a sequence of junction sets \(E_{n}\) with \(E=\bigcup_{n=1}^{\infty}E_{n}\). The main results of the paper under review: Let \(X\) be a topological space, \(Y\) be a separable metrisable space and \(f:X\rightarrow Y\) be a quasi-continuous function. Then the discontinuity point set \(D(f)\) of the function \(f\) is \(\sigma\)-junction set. Let \(X\) be a hereditarily normal space and \(E \subseteq X\). Then \(E\) is a discontinuity point set of some quasi-continuous function \(f:X\rightarrow \mathbb{R}\) if and only if \(E\) is a \(\sigma\)-junction set. In the last theorem of this paper the author proves that if \(X\) is a perfectly normal hereditary quasi-separable (or simple, hereditary separable) Frechet-Uryson space then a subset \(E\) of \(X\) is a discontinuity point set of some quasi-continuous function \(f:X\rightarrow \mathbb{R}\) if and only if \(E\) is a meager \(F_{\sigma}\)-set. Reviewer: Ryszard Pawlak (Łódź) Cited in 2 Documents MSC: 54C30 Real-valued functions in general topology 54C10 Special maps on topological spaces (open, closed, perfect, etc.) Keywords:quasi-continuous function; junction set; \(\sigma\)-junction set; hereditary separable space; discontinuity point PDFBibTeX XMLCite \textit{O. V. Maslyuchenko}, Bull. Aust. Math. Soc. 75, No. 3, 373--379 (2007; Zbl 1123.54005) Full Text: DOI References: [1] Engelking, General topology (1986) [2] Duszynski, Math. Slovaca 51 pp 469– (2001) [3] Breckenridge, Bull. Inst. Math. Acad. Sinica. 4 pp 191– (1976) [4] Maslyuchenko, General Topology in Banach Spañes pp 147– (2001) [5] Maslyuchenko, Ukrainian Math. Zh. 52 pp 740– (2000) [6] Ewert, Real Anal. Exchange 26 pp 687– (2001) [7] Maslyuchenko, Dopovidi Akad. Nauk. Ukrajini pp 28– (1993) [8] Kostyrko, Math. Slovaca. 30 pp 157– (1980) [9] DOI: 10.2307/1990133 · Zbl 0063.09017 · doi:10.2307/1990133 [10] DOI: 10.2307/2040141 · Zbl 0283.26008 · doi:10.2307/2040141 [11] DOI: 10.1007/BF02528836 · doi:10.1007/BF02528836 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.