×

On the topology of generalized quotients. (English) Zbl 1188.54013

Suppose \(G\) is an commutative semigroup acting on a set \(X\). A generalized quotient is defined as a set of equivalence classes of \((x , g)\) where \(x\) is a point in \(X\) and \(g\) is a point in \(G\) as follows: \((x , g)\) is equivalent to \((y , h)\) iff \(g(x) = h(y)\). Topologies on \(X\) and \(G\) induce a natural topology on the corresponding generalized quotient space. The main result of this paper investigates separation properties of generalized quotient spaces. Here is a sample result:
Theorem. If \(X\) is Hausdorff and every point \(g\) in \(G\) is an open map, then the generalized quotient space is Hausdorff.

MSC:

54H15 Transformation groups and semigroups (topological aspects)
54B15 Quotient spaces, decompositions in general topology
20M30 Representation of semigroups; actions of semigroups on sets
54D55 Sequential spaces
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
PDFBibTeX XMLCite
Full Text: DOI