Dimension of dense subalgebras of $C(X)$

Juan B. Sancho de Salas, Ma. Teresa Sancho de Salas
1989 Proceedings of the American Mathematical Society  
The real spectrum of any R-algebra A is the set of all maximal ideals of A with residue field R, endowed with the initial topology for the functions induced by the elements of A. We prove that a compact metric space X has dimension < n if and only if X is the real spectrum of an algebra of Krull dimension < n ; so that the dimension of X is the minimum of the Krull dimensions of all dense subalgebra of C(X). Moreover, we prove that a compact Hausdorff space X has covering dimension < n if and
more » ... ension < n if and only if every countably generated subalgebra of C(X) is contained in the closure of a subalgebra of Krull dimension < n .
doi:10.1090/s0002-9939-1989-0929426-x fatcat:5igromyhnnfq7dry5x3uikx7uq