Perhaps it's only me, but I can't make any sense of your post: you introduce us to X,T (presumably, X is
a points set and T is a topology on it), then you write , which I guess (guess!) means the product of
top. spaces , though the little pi is the usual notation, within this context, for the fundamental group.
Thus, in fact it seems to be that , whereas T is the heaven knows what topology on this
(The prod. topology? The box topology? Other topology?) . You then take an "origin" ...why
the index is i+1?? And what or who is ? The set is, I presume, the original one indexing the product above...?
Then you define a map from (What is this? A coordinate from the element above, or what?) to , but
you define it on some ....and then you ask to show that is homeom. to its image...?
Please try to be way clearer or, preferably, post a link to the original question...or dismiss this post
if I made a whole mess from something simple.