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 $A=\overline{\mathrm{Int} A}.$ 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-\v{C}ech 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 $C_b(X)$.