Comparing T"(X) with C(X)F (subring and equality conditions)

Determine the conditions on a topological space X under which T"(X) is a subring of C(X)F and under which T"(X) equals C(X)F, where C(X)F denotes the ring of real-valued functions on X continuous everywhere except at finitely many points.

Background

Theorem 5.8 establishes that C(X)F is a subring of T"(X) if and only if X is a nowhere almost P-space.

The authors seek the complementary inclusion and the equality criteria to complete the comparison between these rings.