Topologizability of countable equationally Noetherian algebras (Q384392)

An algebra is said to be topologizable if there exists a non-discrete Hausdorff topology such that all basic operations for the algebra \(\mathcal{A }\) are continuous in this topology. An algebraic system is said to be equationally Noetherian if every system \(S\) of equations has a finite subsystem \(S_{0}\) such that sets of solutions for \(S\) and \(S_{0}\) coincide. In this paper the author proves that an arbitrary equationally Noetherian countable algebra in a countable language is topologizable. Also it is shown that certain of the known statements on topologizability follow as a consequence.