On the topologies on ind-varieties and related irreducibility questions

In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevichʼs topology, but irreducible with respect to Kambayashiʼs topology. Moreover, we give a counter-example of a supposed irreducibility criterion given in Shafarevich (1981) [Sha81], which is different from the counter-example given by Homma in Kambayashi (1996) [Kam96]. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich.