Enveloping action: convergence spaces
In this paper the authors point out that quotient maps are central to the development of partial actions. Hereby, a partial action $\alpha$ of a topological group $G$ on a topological space is subject to certain axioms defined in the past. It is known that $\alpha$ is induced by a so-called enveloping action on a suitable chosen topological space $Y$. The topology on $Y$ is the quotient topology determined by an equivalence relation on $G\times Y$. Since in TOP quotient maps are neither productive nor hereditary certain difficulties are encountered in proving. Thus, the authors discuss a partial action and its enveloping action in the larger category CS of Kent convergence spaces and continuous maps which is cartesian closed and extensional and where fewer constraints are required. By using strongly quotient maps it is shown that they are invariant when passing to the enveloping action space. Furthermore, the enveloping action space may fail to be Hausdorff even when the original space is Hausdorff, but in that case an associated Hausdorff enveloping action space and the extension of quotients are presented.
scientific article; zbMATH DE number 7629498
scientific article