On the resolvability of Lindel\"of-generated and (countable extent)-generated spaces

Given a topological property $P$, we say that the space $X$ is $P$-generated if for any subset $A\subset X$ that is not open in $X$ there is a subspace $Y \subset X$ with property $P$ such that $A\cap Y$ is not open in $Y$. (Of course, in this definition we could replace "open" with "closed".) In this paper we prove the following two results: (1) Every Lindel\"of-generated regular space $X$ satisfying $|X|=\Delta(X)={\omega}_1$ is ${\omega}_1$-resolvable. (2) Any (countable extent)-generated regular space $X$ satisfying $\Delta(X)>{\omega}$ is ${\omega}$-resolvable. These are significant strengthenings of our earlier results from [JSSz] which can be obtained from (1) and (2) by simply omitting the "-generated" part. Moreover, the second result improves a recent result of Filatova and Osipov from [FO] which states that Lindel\"of-generated regular spaces of uncountable dispersion character are 2-resolvable. [FO] Maria A. Filatova, Alexander V. Osipov On resolvability of Lindel\"of generated spaces, arxiv:1712.00803. Siberian Electronic Mathematical Reports, Vol. 14, (2017) pp. 1444-1444. [JSSz] Juh\'asz, Istv\'an; Soukup, Lajos; Szentmikl\'ossy, Zolt\'an, Regular spaces of small extent are ${\omega}$-resolvable. Fund. Math. 228 (2015), no. 1, 27-46.