  This note is purely expository. A subset N of the plane is affine ambient
homogeneous if for each x,y in N there exists an affine transformation taking x
to y and N to itself. The result of D. Repovs, E. V. Scepin and the author on
such subsets is presented, together with discussion, corollaries and
generalizations. At the end some non-elementary corollaries are given
(including a simple proof of the smooth version of the Hilbert-Smith conjecture
on topological groups). Most part of the text is accessible to undergraduates
familiar with the notion of continuity. The text could be an interesting easy
reading for mature mathematicians.
