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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.

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