  Haver's near-selection theorem deals with approximate selections of Hausdorff
continuous CE-valued mappings defined on $\sigma$-compact metrizable
$C$-spaces. In the present paper, we extend this theorem to all paracompact
$C$-spaces. The technique developed to achieve this generalisation is based on
oriented simplicial complexes. This approach makes not only a considerable
simplification in the proof but is also successful in generalising the special
case of compact metric $C$-spaces to all paracompact finite $C$-spaces.
