All maps in the lemma hypothesis are continuous.
This is a lemma from Ribes's Profinite Groups. In this proof there is a claim that I can't see why is true. Lets suppose that all topological spaces are groups. So, we are in the context of topological groups and all the maps are continuous homomorphisms.
Why is $\tilde{X}_i$ a group? I don't know if I'm doing some stupid mistake, but I cannot what ensure, for example, that $\tilde{X}_i$ has identity.
I know if I suppose $\tilde{X}_i$ only a topological space, I will get the result for groups, but I'm looking only to the context of groups so I would like to know a proof involving the group properties.
I appreciate any help!