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abstract: '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.'
address: 'Department of Mathematics, Faculty of Science, University of Malta, Msida MSD 2080, Malta'
author:
- Valentin Gutev
title: 'Near-selection theorems for $C$-spaces'
---

Introduction
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A space $X$ has property $C$, or is a *$C$-space*, if for any sequence $\{\mathscr{U} _n:n<\omega\}$ of open covers of $X$ there exists a sequence $\{\mathscr{V} _n:n<\omega\}$ of pairwise disjoint families of open subsets of $X$ such that each $\mathscr{V} _n$ refines $\mathscr{U} _n$ and $\bigcup_{n<\omega}\mathscr{V} _n$ is a cover of $X$. The $C$-space property was originally defined by W.Haver [@haver:1974] for compact metric spaces, subsequently Addis and Gresham [@addis-gresham:78] reformulated Haver’s definition for arbitrary spaces. It should be remarked that a $C$-space is paracompact if and only if it is countably paracompact and normal, see e.g. [@MR2352366 Proposition 1.3]. Every 