| Summary: | We present some lifting theorems for continuous order-preserving functions on locally and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-<i>C</i>-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions.
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