Extending the discrete finite chain condition

AbstractEstablished results show that every zero dimensional, first countable, locally compact space X can be densely embedded in a pseudocompact space Z that is also zero dimensional, first countable and locally compact. the construction relies on a maximal almost disjoint collection of open sets. Because zero dimensionality is present, these sets can be chosen to be clopen and Z is regular as a consequence. Moreover, the set Z − X is closed discrete.Without zero dimensionality, it is still possible to use this mad family approach and an extension Y of X can be constructed where every discrete collection of open sets is finite. In general, the set Y − X need not be closed discrete. This is to be expected; after all, there exist spaces for which no first countable pseudocompactification can be produced by adding a closed discrete set