Not-separation axioms (Q2820413)

In studying so called weakly \(P\) properties in topological spaces the author was led to a property he called not-\(T_0\). A space \((X,\tau )\) is said to be not-\(T_0\) if there exist distinct elements \(x\) and \(y\) in \(X\) such that every open set containing one of \(x\) and \(y\) necessarily contains both of \(x\) and \(y\). In a similiar manner, the properties not-\(T_1\), not-\(T_2\), not-\(R_0\) and not-\(R_1\) can subsequently be defined.NEWLINENEWLINEThe author proves elementary general facts about these properties and considers their behaviour in the case of forming products. As a typical result we mention the following: Let \((X_{\alpha},\tau_{\alpha})\) be a space for each \(\alpha\) and let \(X=\prod\limits_\alpha {X_\alpha }\) carry the product topology \(\tau\). Then \((X,\tau )\) is not-\(T_0\) if there is a \(\beta\) such that \((X_{\beta},\tau_{\beta})\) is not-\(T_0\).