Compactifications of locally compact spaces with zero-dimensional remainder

Baayen, P.C.; Mill, Jan van

doi:10.1016/0016-660x(78)90057-0

Cited by 2 publications

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“…Let ^ be an arbitrary family of sets. Following [1] we denote by \f% the family of all finite unions of sets in *$ and by A^ the family of all finite intersections of sets in S\ Then V A^ = A V ^ is the ring generated by *3".…”

Section: Preliminary Resultsmentioning

confidence: 99%

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Regular Wallman compactifications of rim-compact spaces

Njåstad

1981

J Aust Math Soc A

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A compact Hausdorff space is regular Wallman if it possesses a separating ring of regular closed sets, an j-ring. It was proved by P. C. Baayen and J. van Mill [General Topology and Appl. 9 (1978), [125][126][127][128][129] that if a locally compact Hausdorff space possesses an J-ring, then every Hausdorff compactification with zero-dimensional remainder is regular Wallman.In this paper the reasoning leading to this result is modified to work in a more general setting. Let aX be a Hausdorff compactification of a space X, and let Q a be the family of those closed sets in aX whose boundaries are contained in A". A main result is the following: If (2 T n X contains an j-ring for some Hausdorff compactification yX, then every larger Hausdorff compactification aX for which Q a n (aX -X) is a base for the closed sets on aX -X, is regular Wallman. Various consequences concerning compactifications of a class of rim-compact spaces (called totally rim-compact spaces) are discussed.1980 Mathematics subject classification (Amer. Math. Soc.): 54 D 35.

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Section: Preliminary Resultsmentioning

confidence: 99%

“…(See [11].) In [1] P. C. Baayen and J. van Mill discuss conditions for a Hausdorff compactification of a completely regular space to be regular Wallman. A main result is the following: If a locally compact space possesses an s-ring, then every Hausdorff compactification with zero-dimensional remainder is regular Wallman.…”

Section: Introductionmentioning

confidence: 99%

Regular Wallman compactifications of rim-compact spaces

Njåstad

1981

J Aust Math Soc A

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Richard M. Martin †

Hiż¹

1986

Kant-Studien

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Richard M. Martin f Richard M. Martin, one of the leading analytic philosophers of our times, passed away November 22, 1985. He published profusely, contributed to Kant-Studien. His fifteenth book will appear within a year.Richard M. Martin considered logic to be a fundamental philosophical discipline. He was very proficient in the modern logical techniques but classical in the understanding of the problems of philosophy. Logic is to help to formulate and clarify the problems and the Statements of philosophy. His were the modern masters who continued the classical philosophical thought: Frege, Peirce, Whitehead, Tarski, Carnap. In Martin's works you do not find trivia nor pur technicalia; his every effort is directed toward establishing of a System of great generality and evidence in which all our knowledge can be elucidated. For him "most problems of philosophical analysis hinge at some point on matters of logic. Of course 'logic' must be construed broadly for this claim to have cogency". As all his masters, he defended the classical concept of truth and kept it apart from its illicit Surrogates. His logic was the two-valued classical logic. And he tried to keep logic to its undoubtful basic components. The first Order logic -why to go higher? -was in style of his metaphysical monism. To the logic of sentential connectives and quantifiers he added metalogical concepts like that of designation, denotation, truth and consequence. The resulting System constitutes what he called a good logic. "Not all logics", he wrote, w are on a par and only good logic is an Instrument for keeping us from going wrong, from being too easily misled by every wind of doctrine, from being brain-washed, from becoming intellectually or philosophically 'custom-shrunk'." He did not accept higher order functional calculi, type theory, set theory, or other artificially constructed Systems; no modal logics, no many valued logics. He wanted a philosophically sound logic which will be always a part of our intellectual activity. He quoted approvingly Whitehead: "Speculative philosophy is the endeavor to frame a coherent, logical, necessary System of general ideas in terms of which every element of our experience can be interpreted. It will be observed that logical notions must themselves find their places in the scheme of philosophical notions."If the first order logic is to be the only foundation and we have only one kind of variables, then we have only one ränge of those variables. Therefore, we must reduce the apparent variety of kinds of entities to this single ränge. There are two kinds of reductionism. One is trying to show that all entities are constructions from the fundamental kind. The other reductionism shows that all entities are particular cases of the fundamental ones. Martin taught us how to practice the second type of reductionism. He chose events äs the basic, maybe the most general kind of beings. He presented "a clear cut ontology of events and events only." Other entities, physical things, acts, Brought to you by |

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Compactifications of locally compact spaces with zero-dimensional remainder

Cited by 2 publications

References 10 publications

Regular Wallman compactifications of rim-compact spaces

Regular Wallman compactifications of rim-compact spaces

Richard M. Martin †

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