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I found HERE
A space is called screened if every open covering has a $\sigma$-disjoint open refinement
Do you think this is equivalent to weakly Lindelof: every open covering has a $\sigma$-disjoint subcover ?
That page refers to
D.K. Burke, "Covering properties" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) Chapt. 9; pp. 347–422
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Screened is not equivalent to weakly Lindelof. An uncountable discrete space is screened: indeed every open cover has a disjoint open refinement consisting of singletons. But not every open cover has a sigma-disjoint subcover.
Let $X$ be our uncountable set. Choose one distinguished point $x_0$. Let us consider the open cover $\mathcal U := \{\;\{x_0,x\}\;: x \in X \setminus \{x_0\}\}$ made up of doubletons. A disjoint subfamliy of $\mathcal U$ can only have one set in it, so a sigma-disjoint subfamily of $\mathcal U$ can be at most countable.