Finite unions of locally nice spaces (Q1183633)

Let \(P\) be a ``property''; that is, a class of topological spaces. Then \(P\) is localizable if whenever \(x\in X\in P\), \(x\) has arbitrarily small neighborhoods (not necessarily open) in \(X\) that has property \(P\). It is shown that, for localized and certain ``nice'' properties \(P\), by a standard ``kernel'' inductive procedure, one can characterize those spaces that are expressible as the finite union of subspaces each having the property \(P\) locally, and determine the least such number. Besides, it is shown that this least number, minus one, behaves ``logarithmically'': it is additive for product spaces. Also, specific examples of such properties are given, and some open questions are raised.