Semicontinuous groups and separation properties
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
... 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.