Higher Dimensional Compacta with Algebraically Closed Function Algebras

Abstract

For a compact Hausdorff space $X$, $C(X)$ denotes the ring of all complex-valued continuous functions on $X$. We say that $C(X)$ is \textit{algebraically closed} if every monic algebraic equation with $C(X)$-coefficients has a root in $C(X)$. Modifying the construction of [2], we show that, for each $m = 1,2, \cdots, \infty$, there exists an $m$-dimensional compact Hausdorff space $X(m)$ such that $C(X(m))$ is algebraically closed.