Clopen realcompactification of a mapping (Q1064568)

For a continuous surjective mapping \(f: X\to Y\), where X and Y are Tychonoff, \(\beta\) f denotes the extension of f over the Čech-Stone compactification and \(\upsilon\) f denotes the extension of f over the Hewitt realcompactification. The main result says that \(\upsilon\) f is perfect and open iff f satisfies the following conditions: (1) f(U)\(\subset Int cl f(U)\) for every open set \(U\subset X\), (2) \(cl_{\beta X}f^{-1}(F)=(\beta f)^{-1}(cl_{\beta Y}F)\) for every regular open set \(F\subset Y\), (3) \(\cap \{clf(Z_ n):\) \(n<\omega \}=\emptyset\) whenever \(\{Z_ n: n<\omega \}\) is a decreasing sequence of zero-sets of X with empty intersection. This improves the result of \textit{T. Ishii} [Proc. Japan Acad. 50, 39-43 (1974; Zbl 0299.54004)] which says that if f is quasi-perfect and open, then \(\upsilon\) f is perfect and open.