Ideal resolvability. (English) Zbl 0955.54001
A nonempty topological space is resolvable if it contains two disjoint dense subsets. Let \(\mathcal I\) be an ideal of sets in a topological space \((X,\tau)\). A set \(A\subseteq X\) is \(\mathcal I\)-dense if for every \(x\in X\) and every open neighbourhood \(U\) of \(x\), \(U\cap A\notin\mathcal I\). A topological space is \(\mathcal I\)-resolvable if it contains two disjoint \(\mathcal I\)-dense subsets. It is maximal \(\mathcal I\)-resolvable if it is \(\mathcal I\)-resolvable and no strictly finer topology is \(\mathcal I\)-resolvable. These and related notions are investigated in the paper. In particular, the resolvability of the density topology is proved.
Reviewer: Miroslav Repický (Košice)
MSC:
54A05 | Topological spaces and generalizations (closure spaces, etc.) |
54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |
28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
54G99 | Peculiar topological spaces |