Ambient homogeneity (in Russian)

TL;DR

This paper provides an expository overview of affine ambient homogeneous subsets in the plane, discussing key results, corollaries, and generalizations, including a proof related to the Hilbert-Smith conjecture.

Contribution

It presents a clear exposition of known results on affine ambient homogeneity and introduces some non-elementary corollaries and generalizations.

Findings

Characterization of affine ambient homogeneous subsets

Discussion of corollaries and generalizations

A simple proof of the smooth version of the Hilbert-Smith conjecture

Abstract

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.