\(P\)-spaces and an unconditional closed graph theorem. (English) Zbl 1196.26004
For a completely regular space \(X\) let \(C(X)\), \(U(X)\) and \(B_1(X)\) denote the sets of all respectively continuous real-valued functions, real-valued functions with a closed graph and real-valued functions of the first Baire class. The main result of this paper asserts that \(X\) is a \(P\)-space (that is, every \(G_{\delta}\)-subset of \(X\) is open) if and only if \(U(X)=C(X)\) if and only if \(U(X)=B_1(X)\). Some immediate consequences of this result are presented. In particular, if \(X\) is perfectly normal or first countable or locally compact, then there exist discontinuous functions on \(X\) with a closed graph.
Reviewer: Jorge Picado (Coimbra)
MSC:
| 26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
| 54C30 | Real-valued functions in general topology |
| 54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
| 54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |
Keywords:
\(P\)-spaces; continuous real-valued functions; closed graph real-valued functions; Baire functions; points of discontinuityReferences:
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