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# Symmetric sequence subspaces of $C(\alpha)$, II

Published:1999-04-01
Printed: Apr 1999
• Denny H. Leung
• Wee-Kee Tang
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## Abstract

If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha)$ be the space of all continuous real-valued functions defined on the ordinal interval $[0, \alpha]$. We characterize the symmetric sequence spaces which embed into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy $(E_\alpha)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, $E_\alpha$ embeds into $C(\omega^{\omega^\alpha})$, but does not embed into $C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.
 MSC Classifications: 03E13 - unknown classification 03E1303E15 - Descriptive set theory [See also 28A05, 54H05] 46B03 - Isomorphic theory (including renorming) of Banach spaces 46B45 - Banach sequence spaces [See also 46A45] 46E15 - Banach spaces of continuous, differentiable or analytic functions 54G12 - Scattered spaces