Regular spaces of small extent are -resolvable
Tom 228 / 2015
Streszczenie
We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies \varDelta (X)>\operatorname {\rm e}(X) is {\omega }-resolvable. Here \varDelta (X), the dispersion character of X, is the smallest size of a non-empty open set in X, and \operatorname {\rm e}(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are {\omega }-resolvable.
We also prove that any regular Lindelöf space X with |X|=\varDelta (X)=\omega _1 is even {\omega _1}-resolvable. The question whether regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.
