Question 6: For
example, if one of them is compact, then the other is as well; if
one of them is connected, then the other is as well; if
one of them is Hausdorff, then the other is as well;
their homotopy & ________ will coincide.

Question 7: For some purposes, the homeomorphism group happens to be too
big, but by means of the ________ relation, one can reduce this group
to the mapping class group.