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Compatible extensions of ideals. (English) Zbl 0818.54002
Summary: An ideal on a nonempty set \(X\) is a collection \({\mathcal I}\) of subsets of \(X\) which is closed with regard to finite union and to inclusion. If we assign on \(X\) an ideal \({\mathcal I}\) and a topology \(\tau\), we say that the set \(A\subseteq X\) belongs locally to \({\mathcal I}\), if for every \(a\in A\) there exists a neighborhood \(U\in \tau\) such that \(a\in A\cap U\in {\mathcal I}\); we say that the ideal \({\mathcal I}\) is compatible with the topology \(\tau\) if all sets locally belonging to \({\mathcal I}\) belong to \({\mathcal I}\). It should be noted, that in each topological space compatible ideals are the ideal of nowhere dense sets and the ideal of the sets of first category (the Banach category theorem).
In this paper is characterized the ideal of nowhere dense sets, it is shown that each ideal can be extended canonically to a compatible ideal, and a generalization of the Banach category theorem is proved. The special case of the compatible extension of the ideal of finite subsets is also studied and it is characterized (in a \(T_ 1\) space) as the least ideal on \(X\) containing the ideal of scattered subsets and the ideal of nowhere dense sets.

54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)