Preservation of some covering type properties by linear homeomorphisms of function spaces

An old question of Arhangel'skii asks if (a) the Lindelof property (b) the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space C_p(X) of continuous functions on X equipped with the pointwise topology. A celebrated Velichko's theorem [Topol. Appl. 89, (1998)] provides the affirmative answer to part (a) of this question for linear homeomorphisms of C_p(X)-spaces. In this talk I will show that a similar result (for linear homeomorphisms) holds for part (b). The proof is based on a new method that can also be applied to prove analogous theorems for other, related covering type properties.