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The discontinuity point sets of quasi-continuous functions. (English) Zbl 1123.54005

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.

MSC:

54C30 Real-valued functions in general topology
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
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References:

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[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
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