On homeomorphisms of Cantor cubes

The Cantor cube is an uncountable degree of the Cantor set. Thus, the Cantor cube of continual weight can be represented as a set of Boolean functions on a segment with a topology of pointwise convergence. The main result of the speaker is to prove the strong homogeneity of Cantor cubes. In particular, it follows that the homeomorphism between any compact subsets of a Cantor cube having a weight less than the cube itself extends to an ambient homeomorphism of the entire cube.