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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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