# General affine group is not conjugacy-closed in self-homeomorphism group

## Statement

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.

## Proof

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 .