On \(\theta\)-\(C\)-open sets (Q1187582)

For a space \(X\) and \(A\subseteq X\), recall that \(\text{cl}_ \theta A=\{x\in X\): every closed neighborhood of \(x\) meets \(A\}\). A function \(f:X\to Y\), where \(X\) and \(Y\) are spaces, is \(\theta\)-\(C\)-continuous if for every closed \(A\subset Y\), \(f^ \leftarrow[A]=\text{cl}_ \theta B\) for some \(B\subseteq X\). Properties of \(\theta\)-\(C\)-continuous functions are developed, and applications to \(H\)-closed spaces are presented.