Finiteness at infinity
P. A. Firby
- 01 Dec 1971
- Vol. 17, Iss: 4, pp 299-304
TL;DR: In this paper, the authors give necessary and sufficient conditions for a Tychonoff topological space to be finite at infinity, and they restrict their attention to locally compact, locally compact topological spaces.
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Abstract: If X is a Tychonoff topological space, and if β X is the Stone-Cech compactification of X , then β X \ X will denote the complement of X in β X . If A is a subset of X , then cl [ A : X ] will denote the closure of A in X , and int [ A : X ] will denote the interior of A in X . In Isbell (( 3 ), p. 119) a property of β X \ X is called a property which X has at infinity , and it is the aim of this paper to give necessary and sufficient conditions for X to be finite at infinity. Since β X is T 1 we can say that if X is finite at infinity, then β X \ X is closed in β X . So we lose nothing by restricting our attention to locally compact, Tychonoff spaces, and for the remainder of the paper X will denote such a space.
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Gödel mathematics versus Hilbert mathematics. I The Gödel incompleteness (1931) statement: axiom or theorem?
TL;DR: In this article , the authors focus on the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics.
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Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s 'Completeness Paper' (1930)
TL;DR: In this article , the authors considered the negation of the completeness theorem as a necessary condition for the Gödel incompleteness statement to be a theorem, just as the statement itself to be an axiom.
Finite-point order compactifications
Tom Richmond
- 01 Nov 1987
TL;DR: In this paper, it was shown that if an ordered topological space X has an m-point and an n-point order compactification, then X has a k-point-order compactification for each integer k between m and n.
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