Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

A map $f:X\to Y$ between topological spaces is called weakly discontinuous if each subspace $A\subset X$ contains an open dense subspace $U\subset A$ such that the restriction $f|U$ is continuous. A bijective map $f:X\to Y$ between topological spaces is called a weak homeomorphism if $f$ and $f^{-1}$ are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindel\"of number, dimension. Also we classify infinite zero-dimensional $\sigma$-Polish metrizable spaces up to a weak homeomorphism and prove that any such space $X$ is weakly homeomorphic to one of 9 spaces: $\omega$, $2^\omega$, $\mathbb N^\omega$, $\mathbb Q$, $\mathbb Q\oplus 2^\omega$, $\mathbb Q\times 2^\omega$, $\mathbb Q\oplus\mathbb N^\omega$, $(\mathbb Q\times 2^\omega)\oplus\mathbb N^\omega$, $\mathbb Q\times\mathbb N^\omega$.