Complements of Intersections in Constructive Mathematics
We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or fr...
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| Published in | Mathematical logic quarterly Vol. 40; no. 1; pp. 35 - 43 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin
WILEY-VCH Verlag Berlin GmbH
1994
WILEY‐VCH Verlag Berlin GmbH |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0942-5616 1521-3870 |
| DOI | 10.1002/malq.19940400106 |
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| Abstract | We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and the completeness of metric spaces.
Mathematics Subject Classification: 03F65, 46S30. |
|---|---|
| AbstractList | We examine, from a constructive perspective, the relation between the complements of
S, T
, and
S
∩
T
in
X
, where
X
is either a metric space or a normed linear space. The fundamental question addressed is: If
x
is distinct from each element of
S
∩
T
, if
s
ϵ
S
, and if
t
ϵ
T
, is
x
distinct from
s
or from
t
? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and the completeness of metric spaces.
Mathematics Subject Classification:
03F65, 46S30. We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and the completeness of metric spaces. Mathematics Subject Classification: 03F65, 46S30. |
| Author | Ishihara, Hajime Bridges, Douglas S. |
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| Cites_doi | 10.1007/BF01188684 10.1007/978-3-642-61667-9 10.1017/CBO9780511565663 |
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| Copyright | Copyright © 1994 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim |
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| References | Bridges, D., and F. Richman, Varieties of constructive mathematics. London Math. Soc. Lecture Notes 97, Cambridge University Press, Cambridge 1987. Bishop, E., and D. Bridges, Constructive Analysis. Springer-Verlag, Berlin-Heidelberg-New York 1985. Bishop, E., Foundations of Constructive Analysis. McGraw-Hill, New York 1967. Bridges, D., and H. Ishihara, Linear mappings are fairly well-behaved. Archiv Math. 54 (1990), 558-562. 1987 1986 1985 1990; 54 1967 e_1_2_1_6_2 e_1_2_1_4_2 Bishop E. (e_1_2_1_2_2) 1967 e_1_2_1_5_2 e_1_2_1_3_2 |
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| Snippet | We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear... We examine, from a constructive perspective, the relation between the complements of S, T , and S ∩ T in X , where X is either a metric space or a normed... |
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| SubjectTerms | Complements of intersections Constructive mathematics Intersect sharply Markov's principle Markovian counterexample Metric complement |
| Title | Complements of Intersections in Constructive Mathematics |
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