Topological aspects of order in C(X) 

In this paper we consider the relationship between order and topology in the vector lattice C_b(X) of all bounded continuous
functions on a Hausdorff space X. We prove that the restriction of f\in C_b(X) to a closed set A induces an order continuous 
operator iff A=Cl(Int A). This result enables us to easily characterize bands and projection bands in C_0(X) and C_b(X) 
through the one-point compactification and the Stone-Cech compactification of X, respectively. With these characterizations we
describe order complete C_0(X) and C_b(X)-spaces in terms of extremally disconnected spaces. Our results serve us to solve 
an open question on lifting un-convergence in the case of C_0(X) and C_b(X).