On the topologies on ind-varieties and related irreducibility questions

In the literature there are two ways of endowing an a ne indvariety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of a ne ind-varieties where these two topologies di er. We give an example of an a ne ind-variety that is reducible with respect to Shafarevich's topology, but irreducible with respect to Kambayashi's topology. Moreover, we give a counter-example of a supposed irreducibility criterion given in [Sha81] which is di erent from the counter-example given by Homma in [Kam96]. We nish the paper with an irreducibility criterion similar to the one given by Shafarevich.