  In this paper, we will characterize those sets, over which every irreducible
complete Nevanlinna-Pick space enjoys that its multiplier and supremum norms
coincide. Moreover, we will prove that, if there exists an irreducible complete
Nevanlinna-Pick space of holomorphic functions on a reduced complex space $X$
whose multiplier algebra is isometrically equal to the algebra of bounded
holomorphic functions (we will say that such a space is of Hardy type in this
paper), then $X$ must be a Riemann surface.
