On resolvability of topological spaces
A space \(X\) is \(\kappa\)-resolvable if it can be represented as the union of \(\kappa\)-many mutually disjoint dense subsets. It is maximally resolvable if it is \(\Delta(X)\)-resolvable, where \(\Delta(X)\) is the dispersion character of \(X\). The pseudospread, \(\text{ps}(X)\), (respectively, pseudoextent, \(\text{pext}(X)\)) of a space \(X\) is the minimum cardinal \(\tau^{+}\) such that \(X\) contains no discrete (respectively, closed discrete) subset of cardinality \(\tau\). This paper considers resolvability of spaces whose pseudoextent or pseudospread is less than their dispersion character. The main theorems are as follows: 1) If \(\Delta(X)> \text{ps}(X)\) then \(X\) is maximally resolvable; 2) If \(X\) has a \(\pi\)-network of regular closed sets and \(\Delta(X)> \text{pext}(X)\), then \(X\) is \(\omega\)-resolvable; 3) Under \(V=L\), a dense-in-itself Baire space is \(\omega\)-resolvable. Interesting and immediate corollaries to the first two results are that assuming the negation of the Continuum Hypothesis, a hereditarily Lindelöf connected Hausdorff space is maximally resolvable and a connected regular Lindelöf space is \(\omega\)-resolvable.
scientific article; zbMATH DE number 1891198