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Compactness with respect to an ideal. (English) Zbl 0741.54001
This paper continues and improves the work of R. L. Newcomb[Topologies which are compact modulo an ideal, Ph.D. Dissertation, University of California at Santa Barbara]; see [D. V. Rančin, Sov. Math., Dokl. 13, 193-197 (1972), translation from Dokl. Akad. Nauk SSSR 202, 761-764 (1972; Zbl 0254.54023)] as well. Fix a set \(X\), a topology \(\tau\) on \(X\) and an ideal \({\mathcal I}\) in the power-set of \(X\). The authors call a subset \(A\) of \(X\) \({\mathcal I}\)-compact if for each open cover \(\gamma\) of \(A\), there is a finite subset \(Q_ 1,\dots,Q_ k\) of \(\gamma\) with \(A\backslash (\cup Q_ i)\in {\mathcal I}\). (One can think of the members of \({\mathcal I}\) as ‘small’.) Compactness itself emerges from this in two cases, if \({\mathcal I}=\{\emptyset\}\) or if \({\mathcal I}\) is the ideal of finite sets.
The authors discuss \({\mathcal I}\)-compactness for specific ideals, and more generally, they discover its permanence properties. For example, they describe quasi-\(H\)-closedness, QHC for short. (Recall that \((X,\tau)\) is QHC if for each cover \(\gamma\) of \(X\) there is a finite subset \(Q_ 1,\dots,Q_ k\) of \(\gamma\) such that \(X=\bigcup \hbox{cl}(Q_ i)\).)
Theorem 1.4: If \((X,\tau)\) is \({\mathcal I}\)-compact and \(\tau\cap {\mathcal I}=\{\emptyset\}\) then \((X,\tau)\) is QHC. Conversely, if every nowhere- dense set belongs to \({\mathcal I}\) and \((X,\tau)\) is QHC, then \((X,\tau)\) is \({\mathcal I}\)-compact.
In extending most of the usual compactness theorems, the authors sometimes need the topology \(\tau^*=\tau^*({\mathcal I})\), based on \(\beta(\tau,{\mathcal I})=\{T\backslash I: T\in \tau \& I\in{\mathcal I}\}\), see [P. Samuels, J. London Math. Soc., II. Ser. 10, 409-416 (1975; Zbl 0303.54001)]. In particular,
— they characterize \({\mathcal I}\)-compactness by a sort of finite intersection property, by the existence of accumulation points for filter (bases) in \({\mathcal P}(X)\backslash{\mathcal I}\), and by the existence of limits for ultrafilters in \({\mathcal P}(X)\backslash{\mathcal I}\),
— they investigate the preservation of \({\mathcal I}\)-compactness under continuity,
— they obtain a Tychonov product theorem,
— they derive heredity theorems for \(\tau^*\)-closed subsets of \({\mathcal I}\)-compact sets,
— they provide many other results and examples.

54A05 Topological spaces and generalizations (closure spaces, etc.)
54D30 Compactness