General affine group is not conjugacy-closed in self-homeomorphism group
The general affine group of all invertible affine transformations over the field of real numbers, is not conjugacy-closed in the self-homeomorphism group of . In other words, there are affine transformations which are not conjugate in but are n
In particular, this also yields that the self-diffeomorphism group of , is not conjugacy-closed in the self-homeomorphism group, because general affine group is conjugacy-closed in self-diffeomorphism group.
Put . Then the transformations and , are conjugate in the self-homeomorphism group if are both between and . The conjugating homeomorphism is the map:
and is defined to be at .
Note that this conjugating homeomorphism is not a diffeomorphism.
This same example can be used for the case , by taking scalar matrices for and .