Near-selection theorems for C-spaces

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 show that it is valid for all paracompact $C$-spaces, which answers in the affirmative a question raised by V. Valov. The technique developed to achieve this result is based on oriented simplicial complexes. It makes not only a considerable simplification in the proof but is also successful in generalising an alternative near-selection theorem for compact metric $C$-spaces to all paracompact finite $C$-spaces.