On generalized pre-closure spaces and separation for some special types of functions (Q2881396)

The results of the present paper include the following:NEWLINENEWLINE Result 1: Let \((X,p\,\text{Cl}_X)\) and \((Y,p\,\text{Cl}_Y)\) be generalized pre-closure spaces. Let \(f: X\to Y\). If \(f\) is pre-closure-preserving, then \(f\) is non-pre-separating.NEWLINENEWLINE Result 2: Let \(f:(X,p\,\text{Cl})\to (Y,p\,\text{Cl})\) and \(g:(Y, p\,\text{Cl})\to (Z,p\,\text{Cl})\) be precontinuous functions. Then \(g\circ f: X\to Z\) is precontinuous.NEWLINENEWLINE Result 3: Let \((X,p\,\text{Cl})\) and \((Y,p\,\text{Cl})\) be generalized pre-closure spaces with pre-grounded pre-isotonic \(p\,\text{Cl}\) and let \(f: (X,p\,\text{Cl})\to (Y,p\,\text{Cl})\) be a precontinuous surjection. If \(X\) is strongly preconnected, then \(Y\) is strongly preconnected.