Hausdorff properties of topological algebras.
A topological space \(X\) is \(T_0\) if for each distinct points \(a,b\) of \(X\) there is a closed subset containing just one of the points. \(X\) is \(T_1\) if for each \(a\in X\) the singleton \{\(a\)\} is closed. \(X\) is \(T_2\) if for each \(a\in X\) the intersection of the closures of the neighborhoods of \(a\) is \{\(a\)\}. It was shown by W. Taylor and H. P. Gumm that if \(\mathcal V\) is a \(k\)-permutable variety then any topological algebra in \(\mathcal V\) satisfies \(T_0 \Rightarrow T_1\) and for \(k=3\) also \(T_0 \Rightarrow T_2\). J. Coleman defined recently new separation conditions \(H_j\) called \(j\)-step Hausdorffness for each \(j\geq 1\) and proved that if \(\mathcal V\) is \(k\)-permutable then each topological algebra in \(\mathcal V\) satisfies \(T_0\Rightarrow H_{k-2}\). The authors proved that if \(\mathcal V\) is congruence modular and \(k\)-permutable then every topological algebra of \(\mathcal V\) satisfies \(T_0\Rightarrow T_2\), the same for a weakly regular variety. If \(\mathcal V\) is \(k\)-permutable then \(T_0\Rightarrow H_{\left [{k\over 2}\right ]}\) but not \(T_0\Rightarrow H_{{\left [{k\over 2}\right ]}-1}\) for topological algebras of \(\mathcal V\). Moreover, the class of varieties where topological algebras satisfy \(T_0\Rightarrow T_i\) or \(T_0\Rightarrow H_j\) is definable by a Mal'tsev condition.
scientific article; zbMATH DE number 2158785