Topological equivalents to \(n\)-permutability (Q1272150)

A topological algebra is a pair \((A,\tau)\) where \(A\) is an algebra and \(\tau \) is a topology on \(A\) such that all basic operations on \(A\) are continuous w.r.t. the topology \(\tau \). \(A\) satisfies the \(R_0\) separation axiom if for each \(a\in U\), \(U\) open, also \(\text{cl}(a)\subseteq U\). The following interesting main theorem characterizing \(n\)-permutable varieties topologically is proved: For a variety \(\mathcal V\) the following are equivalent: (1) \(\mathcal V\) is \(n\)-permutable for some \(n\). (2) Every topological \(\mathcal V\)-algebra is \(R_0\). (3) Every \(T_0\) topological \(\mathcal V\)-algebra is \(T_1\) and sober. (4) Every \(T_0\) topological \(\mathcal V\)-algebra is \(T_1\).