New topologies from old via ideals.

*(English)*Zbl 0723.54005Let (X,\(\tau\)) be a topological space, with \({\mathcal I}\) an ideal of subsets of X. Then \(\beta\) (\({\mathcal I}):=\{U\setminus I:\) \(I\in {\mathcal I}\}\) is a basis of open sets for a finer topology \(\tau\) (\({\mathcal I})\) on X. The authors explore this time-honored method of refining topologies, surveying past results, proving some new results, and improving on old ones. The treatment is tutorial in nature, and includes many examples. As the authors suggest, the paper is suitable for use as a supplement to a general topology course.

The first three sections of the paper deal with the closure and derived set operators for \(\tau\) (\({\mathcal I})\). (Actually this topology is introduced using a closure operator.) In Section 4, O. Njåstad’s notion of “compatibility” is introduced: The topology \(\tau\) is compatible with the ideal \({\mathcal I}\) (\(\tau\sim {\mathcal I})\) if \(A\in {\mathcal I}\) whenever it is the case that for all \(x\in A\), \(U\cap A\in {\mathcal I}\) for some neighborhood U of x. The authors show that \(\tau\sim {\mathcal I}\) whenver \({\mathcal I}\) is the ideal of \(\tau\)-nowhere dense subsets of X, and recast the Banach category theorem (that any union of meager open sets is meager) as the statement that \(\tau\sim {\mathcal I}\) whenever \({\mathcal I}\) is the ideal of \({\mathcal T}\)-meager subsets of X. A nice result is that \(\tau\) is a hereditarily Lindelöf topology iff \(\tau\) is compatible with the ideal of countable subsets of X. Also there is Njåstad’s result that \(\beta\) (\({\mathcal I})=\tau ({\mathcal I})\) whenever \(\tau\sim {\mathcal I}.\)

A highlight of Section 5 is G. Freud’s generalization of the Cantor-Bendixson theorem (that any second countable (even hereditarily Lindelöf) space is the union of a perfect subset and a countable subset), namely: If \(\tau\sim {\mathcal I}\) and \({\mathcal I}\) contains the singleton subsets of X, then every \(\tau\) (\({\mathcal I})\)-closed subset is the union of a \(\tau\)-perfect set and a set that is in \({\mathcal I}\). Also the authors prove that \(\tau\sim {\mathcal I}\) and \({\mathcal I}\) contains all the singletons iff all \(\tau\) (\({\mathcal I})\)-scattered subsets of X are in \({\mathcal I}.\)

In Section 6 the authors consider the case when no nonempty \(\tau\)-open set is in \({\mathcal I}\), and show that this condition implies that both \(\tau\) and \(\tau\) (\({\mathcal I})\) have the same semiregularization. The authors also state P. Samuels’ theorem that, under this condition, a function from X to a regular space is continuous with respect to \(\tau\) iff that function is continuous with respect to \(\tau\) (\({\mathcal I}).\)

Finally, in Section 7 there are some applications. One such is the ease with which “anticompact” spaces (those containing no infinite compact subsets) can be produced. For example, if \(\tau\) is a Hausdorff topology compatible with \({\mathcal I}\), and if \({\mathcal I}\) contains all the singleton subsets of X, then \(\tau\) (\({\mathcal I})\) is anticompact. Other applications involving continuity and \(\theta\)-continuity are also given.

The first three sections of the paper deal with the closure and derived set operators for \(\tau\) (\({\mathcal I})\). (Actually this topology is introduced using a closure operator.) In Section 4, O. Njåstad’s notion of “compatibility” is introduced: The topology \(\tau\) is compatible with the ideal \({\mathcal I}\) (\(\tau\sim {\mathcal I})\) if \(A\in {\mathcal I}\) whenever it is the case that for all \(x\in A\), \(U\cap A\in {\mathcal I}\) for some neighborhood U of x. The authors show that \(\tau\sim {\mathcal I}\) whenver \({\mathcal I}\) is the ideal of \(\tau\)-nowhere dense subsets of X, and recast the Banach category theorem (that any union of meager open sets is meager) as the statement that \(\tau\sim {\mathcal I}\) whenever \({\mathcal I}\) is the ideal of \({\mathcal T}\)-meager subsets of X. A nice result is that \(\tau\) is a hereditarily Lindelöf topology iff \(\tau\) is compatible with the ideal of countable subsets of X. Also there is Njåstad’s result that \(\beta\) (\({\mathcal I})=\tau ({\mathcal I})\) whenever \(\tau\sim {\mathcal I}.\)

A highlight of Section 5 is G. Freud’s generalization of the Cantor-Bendixson theorem (that any second countable (even hereditarily Lindelöf) space is the union of a perfect subset and a countable subset), namely: If \(\tau\sim {\mathcal I}\) and \({\mathcal I}\) contains the singleton subsets of X, then every \(\tau\) (\({\mathcal I})\)-closed subset is the union of a \(\tau\)-perfect set and a set that is in \({\mathcal I}\). Also the authors prove that \(\tau\sim {\mathcal I}\) and \({\mathcal I}\) contains all the singletons iff all \(\tau\) (\({\mathcal I})\)-scattered subsets of X are in \({\mathcal I}.\)

In Section 6 the authors consider the case when no nonempty \(\tau\)-open set is in \({\mathcal I}\), and show that this condition implies that both \(\tau\) and \(\tau\) (\({\mathcal I})\) have the same semiregularization. The authors also state P. Samuels’ theorem that, under this condition, a function from X to a regular space is continuous with respect to \(\tau\) iff that function is continuous with respect to \(\tau\) (\({\mathcal I}).\)

Finally, in Section 7 there are some applications. One such is the ease with which “anticompact” spaces (those containing no infinite compact subsets) can be produced. For example, if \(\tau\) is a Hausdorff topology compatible with \({\mathcal I}\), and if \({\mathcal I}\) contains all the singleton subsets of X, then \(\tau\) (\({\mathcal I})\) is anticompact. Other applications involving continuity and \(\theta\)-continuity are also given.

Reviewer: P.Bankston (Milwaukee)

##### MSC:

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