Discontinuous Homomorphisms of \(C(X)\) with \(2^{\aleph_0}>\aleph_2\)

• Published in: arXiv.org
• Main Author: Dumas, Bob A
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.05.2019
• Subjects: Continuity (mathematics); Homomorphisms; Mathematical analysis; Mathematical functions; Power series
• ISSN: 2331-8422

Assume that \(M\) is a c.t.m. 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 \(2^{\aleph_0}=\aleph_3\) and, for any infinite compact Hausdorff space \(X\), there exists a discontinuous homomorphism of \(C(X)\), the algebra of continuous real-valued functions on \(X\). For \(n\in \mathbb N\), If \(M\) contains a simplified \((\omega_1,n)\)-morass, then in the Cohen extension of \(M\) adding \(\aleph_n\) generic reals there exists a discontinuous homomorphism of \(C(X)\), for any infinite compact Hausdorff space \(X\).