\(P\)-spaces and an unconditional closed graph theorem (Q981967)

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.