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

## TL;DR

This paper demonstrates the consistency of the existence of discontinuous homomorphisms of $C(X)$ for any infinite compact Hausdorff space $X$ under certain set-theoretic assumptions involving forcing and morasses.

## Contribution

It constructs a model where $2^{eth_0}=eth_3$ and shows that discontinuous homomorphisms of $C(X)$ exist for all infinite compact Hausdorff spaces, extending previous results.

## Key findings

- Discontinuous homomorphisms of $C(X)$ exist in the constructed model.
- The existence depends on set-theoretic assumptions involving morasses and forcing.
- The results hold for any infinite compact Hausdorff space $X$.

## Abstract

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 $X$.