Locally Constant Functions In The Functionally Countable Subalgebra Of C(X)

Locally Constant Functions In The Functionally Countable Subalgebra Of C(X)

We show that Cc (X ) the subalgebra of C(X ) consisting of elements with countable image and C(X ) share some interesting properties with regard to their rings of quotients. We study c-locally constant functions in Cc (X), relative rings and connections that these rings have together. In particular, it is also shown that Qc(X ), the maximal ring of quotients of Cc(X), is metrically dense in CC [ ] = ∪{Cc (S ) : S ∈ }, where the filter base contains the dense open subsets of X