as a valuation domain versus an cp-space x

Let cc(x) be the functionally countable subalgebra of c(x). is real closed if and only if it is a valuation ring, in this case x is called a csv -space. for each zero dimensional space x, cc(x) is a csv -ring if and only if cc(υx) is a csv -ring. for each x ∈ x rank x is the rank of m^c_x. it is shown that for a proper essential cp-space x with non cp-point η, the rank of cc(x) is the rank of m^c_η and essential cp-space x is a csv -space if and only if it hasfinite rank.