### Semicontinuous groups and separation properties

Ellen Clay, Bradd Clark, Vic Schneider
1992 International Journal of Mathematics and Mathematical Sciences
In 1948, Samuel  pointed out that the intersection of two group topologies need not be a group topology. However, a number of properties that hold for a group topology still hold for a topological space that is an intersection of group topologies. In order to study these properties, we shall describe a class of topologies that can be placed on a group which we call semicontinuous topologies. (We point out here that Fuchs  calls these spaces semitopological groups). One important attribute
more » ... important attribute of topological groups is separation. In particular, a topological group is Hansdorff if and only if the identity is a closed subset. While this is not true for semicontinuous groups, we shall see that an interesting "echo" of this property is true. For each group G we have a bijection inv: G-,G defined by inv (x) x-1. Also for any fixed a E G we have bijections la:G-,G defined by la(x) az and ra:G-G defined by DEFINITION. A semicontinuous gLq. is a group G and a topology r on G making inv, and r a continuous for a E G. Clearly a semicontinuous group is a homogeneous space. Thus a great deal can be determined by considering a basis for the topology at the identity. In a manner imalogous to that found in the theory of topological groups, one can demonstrate the following: PROPOSITION 1. tf (G, r) is a semicontinuous group and Y is a neighborhood base at the identity, then Y satisfies (i) If U, V f, then there exists W /Y such that W C U f U. (ii) If a U and U Y, then there exists V E Y such that Va C U. (iii) If U Y then there exists V (/Y such that V -1 C U. (iv) If U Y and x (/G then there exists V (/Y such that zVz -1 C U. Furthermore, if Y is any collection of subsets of G, each containing the identity, and Y satisfies (i)-(iv) above, then there exists a unique semicontinuous topology on G for which Y is a neighborhood base at the identity.