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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear spaces, absolute retracts, and the compact extension property


Authors: Jos van der Bijl and Jan van Mill
Journal: Proc. Amer. Math. Soc. 104 (1988), 942-952
MSC: Primary 57N17; Secondary 54C20, 54F40, 55M15
MathSciNet review: 964878
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Abstract: We formulate a "partial realization" property and prove that this property is equivalent to the compact extension property. In addition, we prove that a linear space $ L$ has the compact extension property if and only if $ L$ is admissible if and only if $ L$ has the $ \sigma $-compact extension property. This implies that for a $ \sigma $-compact linear space $ L$, the following statements are equivalent: (1) $ L$ is an absolute retract, (2) $ L$ has the compact extension property, and (3) $ L$ is admissible. Finally, we prove that if there exists a linear space which is not an absolute retract then there is an admissible linear space which is not an absolute retract.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0964878-X
PII: S 0002-9939(1988)0964878-X
Keywords: Absolute retract, the compact extension property, linear space, admissible linear space
Article copyright: © Copyright 1988 American Mathematical Society