Topological groups acting on sets (Q1822084)

The authors consider a topological group G and a set X on which G acts, and they study the topology on X in which a set A is open if and only if for every \(x\in A\) there exists a neighbourhood U of the identity in G such that Ux\(\subseteq A\). Among various other elementary statements about this topology they show that it coincides with the final topology coinduced by the evaluation maps \(G\to X: g\mapsto gx\) for \(x\in X\). They seem not to realize that with this topology (G,X) becomes a topological transformation group; instead, they give a number of statements (seemingly at random) which are well-known immediate consequences of this fact. Finally they show that if G is first countable then X has a \(G_{\delta}\)-diagonal. \{Reviewer's remark: Warning: The paper contains several erroneous statements. Corollary 4 is false (just use \(X=G\) with the action of G by left translations to construct counterexamples). Therefore the statement that X is regular (which is claimed to be the main result, and which is true) stays unproved. Corollary 9 is mysterious since, if taken literally, (''Simple groups act transitively''), it is blatantly wrong. The paper contains numerous misprints, grammatical slips, distorted expressions and as a whole is written rather carelessly (for instance there are sentences which lead nowhere: remmants of earlier versions?). One wonders if the article has been refereed; it should not have been printed in this state.\}