Regular spaces of small extent are \(\omega \)-resolvable (Q2928546)

Let \(X\) be a topological space. For a cardinal \(\lambda\), \(X\) is \(\lambda\)-resolvable if it contains \(\lambda\) many mutually disjoint dense subsets. The dispersion \(\Delta(X)\) of \(X\) is the minimum cardinality of non-empty open subsets of \(X\) while the extent e\((X)\) of \(X\) is the supremum of the cardinalities of closed discrete subsets of \(X\). It is shown that if \(X\) is regular with \(\Delta(X)>\text{e}(X)\) then \(X\) is \(\omega\)-resolvable. If, further, \(X\) is Lindelöf and \(|X|=\Delta(X)=\omega_1\) then \(X\) is \(\omega_1\)-resolvable.