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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 \((X,\tau)\).

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D30 Compactness
54D45 Local compactness, \(\sigma\)-compactness
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[1] RANČIN D. V.: Compactness modulo an ideal. Soviet Math. Dokl. 13(1) (1972), 193-197. · Zbl 0254.54023
[2] NEWCOMB R. L.: Topologies Which are Compact Modulo an Ideal. Ph.D. dissertation, Univ. of Cal. at Santa Barbara, 1967.
[3] HAMLETT T. R, JANKOVIČ D.: Compactness with respect to an ideal. Boll. Un. Mat. Ital. B (7) 4 (1990), 849-861. · Zbl 0741.54001
[4] HAMLETT T. R., ROSE D.: Local compactness with respect to an ideal. Kyung Pook Math. J. 32 (1992), 31-43. · Zbl 0767.54019
[5] PORTER J.: On locally H-closed spaces. Proc. London Math. Soc. (3) 20 (1970), 193-204. · Zbl 0189.53404
[6] VAIDYANATHASWAMY R.: Set Topology. Chelsea Publishing Company, New York, 1960. · Zbl 0923.54001
[7] NJÅSTAD O.: Classes of topologies defined by ideals. Matematisk Institutt, Universitetet I Trondheim) · Zbl 0148.16504
[8] NJÅSTAD O.: Remarks on topologies defined by local properties. Det Norske Videnskabs-Akademi, Avh. I Mat. Naturv, Klasse, Ny Serie No. 8 (1966), 1-16. · Zbl 0148.16504
[9] JANKOVIČ D., HAMLETT T. R.: New topologies from old via ideals. Amer. Math. Monthly 97 (1990), 255-310. · Zbl 0723.54005
[10] JANKOVIČ D., HAMLETT T. R.: Compatible extensions of ideals. Boll. Un. Mat. Ital. B (7)) · Zbl 0818.54002
[11] VAIDYANATHASWAMY R.: The localization theory in set-topology. Proc. Indian Acad. Sci. Math. Sci. 20 (1945), 51-61. · Zbl 0061.39308
[12] SEMADENI Z.: Functions with sets of points of discontinuity belonging to a fixed ideal. Fund. Math. LII (1963), 25-39. · Zbl 0146.12302
[13] OXTOBY J. C.: Measure and Category. Springer-Verlag, New York, 1980. · Zbl 0435.28011
[14] SAMUELS P.: A topology formed from a given topology and ideal. J. London Math. Soc. (2) 10 (1975), 409-416. · Zbl 0303.54001
[15] BANKSTON P.: The total negation of a topological property. Illinois J. Math. 23 (1979), 241-252. · Zbl 0405.54003
[16] KELLEY J. T.: General Topology. D. Van Nostrand Company, Inc., Princeton, 1955. · Zbl 0066.16604
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