On the A-invariance of the hereditary Baire property and related results

On the A-invariance of the hereditary Baire property and related results
Abstract: During the talk we will show that if X and Y are first-countable perfect spaces such that the free Abelian topological groups A(X) and A(Y) are topologically isomorphic, then X is a hereditarily Baire space if and only if Y is hereditarily Baire as well. We will also establish that for any Tychonoff space, if there exists a continuous linear surjection of the space C_p(X) onto the space C_p(Y) and the space X is either strongly σ-scattered or has property (κ), then Y also satisfies those properties. Additionally, we will show the following result: if X and Y are Tychonoff spaces in which every closed set has a W-point, and if the free Abelian topological groups A(X) and A(Y) are topologically isomorphic, then X is scattered if and only if Y is scattered. The talk is based on the following preprint, written jointly with Mikołaj Krupski: https://arxiv.org/abs/2608.11115