On minimal Hausdorff and minimal Urysohn functions (Q651295)

The authors develop a new characterization of minimal Hausdorff functions and introduce the concept of minimal Urysohn functions. They also investigate hereditary and productive properties for both minimal Hausdorff and Urysohn functions. A Hausdorff space \(X\) is \textit{minimal Hausdorff} if it has no strictly coarser Hausdorff topology. A continuous function \(f:X\to Y\) is said to be \textit{Hausdorff} (respectively, \textit{Urysohn}) if for every \(x,y\in X\) such that \(x\not=y\) and \(f(x)=f(y)\) there are two neighbourhoods \(U_x\) and \(V_x\) of \(x\) and \(y\) such that \(U_x \bigcap V_y = \emptyset\) (respectively, \(cl_X(U_x) \bigcap cl_X(V_y)=\emptyset\)). A Hausdorff function \(f:(X, \tau(X))\to Y\) is \textit{minimal Hausdorff} (respectively, \textit{minimal Urysohn}) if for any topology \(\sigma\) on \(X\), \(\sigma \not\subset \tau(X)\), \(f^{\prime}:(X,\sigma)\to Y\), \(x\to f(x)\), is not Hausdorff (respectively, Urysohn).