No CrossRef data available.
Article contents
DISCONTINUOUS HOMOMORPHISMS OF 
$C(X)$ WITH 
$2^{\aleph _0}>\aleph _2$
Published online by Cambridge University Press: 15 April 2024
Abstract
Assume that M is a transitive model of 
$ZFC+CH$ containing a simplified 
$(\omega _1,2)$-morass, 
$P\in M$ is the poset adding 
$\aleph _3$ generic reals and G is P-generic over M. In M we construct a function between sets of terms in the forcing language, that interpreted in 
$M[G]$ is an 
$\mathbb R$-linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on 
$\omega $, into the Esterle algebra of formal power series. Therefore it is consistent that 
$2^{\aleph _0}>\aleph _2$ and, for any infinite compact Hausdorff space X, there exists a discontinuous homomorphism of 
$C(X)$, the algebra of continuous real-valued functions on X.
Keywords
Information
- Type
- Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
Dales, H. G., Automatic continuity: A survey. Bulletin of the London Mathematical Society, vol. 10 (1978), pp. 129–183.CrossRefGoogle Scholar
Dales, H. G., A discontinuous homomorphism of C(X). American Journal of Mathematics, vol. 101 (1979), no. 3, pp. 647–734.CrossRefGoogle Scholar
Dumas, B. A., Order-isomorphic
${\eta}_1$
-orderings in Cohen extensions. Annals of Pure and Applied Logic, vol. 158 (2009), pp. 1–22.CrossRefGoogle Scholar
${\eta}_1$
-orderings in Cohen extensions. Annals of Pure and Applied Logic, vol. 158 (2009), pp. 1–22.CrossRefGoogle ScholarDumas, B. A., Gap-2 Morass-definable
${\eta}_1$
-orderings. Mathematical Logic Quarterly, vol. 68 (2022), no. 2, pp. 227–242.CrossRefGoogle Scholar
${\eta}_1$
-orderings. Mathematical Logic Quarterly, vol. 68 (2022), no. 2, pp. 227–242.CrossRefGoogle ScholarErdös, P., Gillman, L., and Henriksen, M., An isomorphism theorem for real closed fields. Annals of Mathematics, vol. 61 (1955), pp. 552–554.CrossRefGoogle Scholar
Esterle, J., Sur l’existence d’un homomorphisme discontinu do C(K). Acta Mathematica Academiae Scientiarum Hungarica, vol. 30 (1977), pp. 113–127.CrossRefGoogle Scholar
Esterle, J., Semi-normes on
$C(K)$
. Proceedings of the London Mathematical Society, vol. 36 (1978), no. 3, pp. 27–45.CrossRefGoogle Scholar
$C(K)$
. Proceedings of the London Mathematical Society, vol. 36 (1978), no. 3, pp. 27–45.CrossRefGoogle ScholarEsterle, J., Injections de semi-groupes divisible dans des algébras de convolution et construction d’homomorphismes discontinus des C(K), Proceedings of the London Mathematical Society, vol. 36 (1978), no. 3, pp. 59–85.CrossRefGoogle Scholar
Hahn, H., Ober die nichtarchimedischen Grössensysteme. Sitzungsberichte Kaiserlichen der Akademie Wissenschaften, Wien, Mathematisch–Naturwissenschaftliche Klasse 116 (Abteilung IIa, 1907), pp. 601–655.Google Scholar
Johnson, B. E., Norming
$C(\varOmega)$
and related algebras. Transactions of the American Mathematical Society, vol. 220 (1976), pp. 37–58.Google Scholar
$C(\varOmega)$
and related algebras. Transactions of the American Mathematical Society, vol. 220 (1976), pp. 37–58.Google ScholarMaclane, S., The universality of power series fields. Bulletin of the American Mathematical Society, vol. 45 (1939), pp. 888–890.CrossRefGoogle Scholar
Szalkai, I., An inductive definition of higher gap simplified morasses. Publicationes Mathematicae Debrecen, vol. 58 (2001), no. 4, pp. 605–634.CrossRefGoogle Scholar
Velleman, D., Simplified morasses, Journal of Symbolic Logic, vol. 49 (1984), pp. 257–271.Google Scholar
Velleman, D., Simplified gap-2 morasses. Annals of Pure and Applied Logic, vol. 34 (1987), pp. 171–208.CrossRefGoogle Scholar
Woodin, H., A discontinuous homomorphism from C(X) without CH. Journal of the London Mathematical Society, vol. s2-48 (1993), no. 2, pp. 299–315.CrossRefGoogle Scholar