Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

We present some lifting theorems for continuous order-preserving functions on locally and σ-compact Hausdorff preordered topological spaces. In particular, we show that a preorder space has multi-utility representation if, only every compact subspace, function can be lifted to the entire space. Such characterization is also presented by introducing property of ≾-C-compatible closed subspaces. The assumption paracompactness used in connection conditions.