The Vietoris modifications of a frame

The many different point-sensitive Vietoris modifications of a topological space have been around for many years. These constructions produce what are often called hyperspaces of a space. The point-free version, which is rather more modern, was introduced in [3] and reworked in [4]. Since then nothing much worth reading has been written about the construction, although [6] is an exception to this. Much of what has been written seems to have been from the rather narrow perspective of squeezing a bit more out of some power domain construction or other. I take the wider view that the constructions are bits of topological gadgetry which are interesting in their own right. Here I set down some of the basic material together with a few observation I have made over the years. As will become clear, these notes are based in part on [3, 4] with the benefit of some 20 years of further development in the use of frames as a tool for studying certain aspects of point set topology. I have also gathered together the contents of various scraps of paper I have had hanging around for a while. I’m not sure there is much new here, but you might find them useful.