Linear spaces, absolute retracts, and the compact extension property

van der Bijl, Jos; van Mill, Jan

doi:10.1090/S0002-9939-1988-0964878-X

Linear spaces, absolute retracts, and the compact extension property
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by Jos van der Bijl and Jan van Mill

Proc. Amer. Math. Soc. 104 (1988), 942-952

DOI: https://doi.org/10.1090/S0002-9939-1988-0964878-X

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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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