On isomorphism classes of $C({\bf 2}^{\mathfrak{m}} \oplus [0, \alpha])$ spaces
Volume 204 / 2009
Fundamenta Mathematicae 204 (2009), 87-95
MSC: Primary 46B03, 46E15; Secondary
03E55.
DOI: 10.4064/fm204-1-5
Abstract
We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces ${\bf 2}^{\mathfrak{m}} \oplus [0, \alpha]$, the topological sums of Cantor cubes ${\bf 2}^{\mathfrak{m}}$, with $\mathfrak{m}$ smaller than the first sequential cardinal, and intervals of ordinal numbers $[0, \alpha]$. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C({\bf 2}^{\mathfrak{m}} \oplus [0, \alpha])$ spaces with $\mathfrak{m} \geq \aleph_{0}$ and $\alpha \geq \omega_1$ are the trivial ones. This result leads to some elementary questions on large cardinals.
