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On one-point $${\mathcal I}$$-compactification and local $${\mathcal I}$$- compactness. (English) Zbl 0767.54020
Summary: An ideal $$\mathcal I$$ on a set $$X$$ is a nonempty subset of the power set $${\mathcal P}(X)$$ which has heredity and is finitely additive. (Local) $$\mathcal I$$-compactness is the natural generalization of (local) compactness, where an $$\mathcal I$$-cover of $$A\subseteq X$$ covers all but an ideal member of $$A$$. If $$\tau$$ is a topology on $$X$$, $$\mathcal I$$ is $$\tau$$-codense if each member of $$\mathcal I$$ is codense in $$(X,\tau)$$ and $$\mathcal I$$ is $$\tau$$- local if each subset $$A\subseteq X$$ locally in $$\mathcal I$$ belongs to $$\mathcal I$$. If $$\mathcal I$$ is $$\tau$$-local, then $$\beta=\{U-I\mid U\in \tau,\;I\in {\mathcal I}\}$$ is a topology. In any case, $$\beta$$ is a basis for a topology $$\tau^*({\mathcal I})$$ finer than $$\tau$$ on $$X$$. It is seen that a Hausdorff space $$(X,\tau)$$ has a one-point Hausdorff $$\mathcal I$$- compactification if and only if each point of $$X$$ has a $$\tau$$-closed $$\mathcal I$$-compact neighbourhood. This condition which is equivalent to $$(X,\tau^*({\mathcal I}))$$ being locally $$\mathcal I$$-compact, properly implies that $$(X,\tau)$$ is locally $$\mathcal I$$-compact. However, the converse is implied by the $$\tau$$-codenseness of $$\mathcal I$$. Further, when $$\mathcal I$$ is $$\tau$$-codense, $$(X,\tau)$$ having a one-point Hausdorff $$\mathcal I$$- compactification implies that $$(X,\tau)$$ is locally $$H$$-closed, i.e. locally $${\mathcal N}(\tau)$$-compact, where $${\mathcal N}(\tau)$$ is the ideal of nowhere dense subsets of $$(X,\tau)$$.

MSC:
 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D30 Compactness 54D45 Local compactness, $$\sigma$$-compactness
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