Symmetrization of closure operators and visibility (Q818919)

This paper describes a canonical method of associating with any finitary closure operator \(\sigma\) on a set \(E\) a finitary closure operator \(\hat\sigma\) such that (i) for \(A \subseteq E\), \(\hat\sigma(A) \supseteq \sigma(A)\), and (ii) \(\hat\sigma\), in the terminology of the paper, is ``symmetric'' (that is, obeys the matroid exchange axiom). This construction is compared to another such canonical method, given in [\textit{A. W. M. Dress} and \textit{W.~Wenzel}, ``Matroidizing set systems -- a new approach to matroid theory'', Appl. Math. Lett. 3, 29--32 (1990; Zbl 0701.05013)], which works only for finite \(E\). \noindent Motivated by examples of closure operators arising in connection with Hadwiger's visibility problem for convex sets in Euclidean spaces, the constuction of \(\hat\sigma\) is studied particularly with respect to closure operators which satisfy a ``visibility'' condition. (These closure operators can be described as being those which are obtained from a pre-order \(\leq\) on \(E\) by letting \(\sigma(A) = \{e \in E : \text{\;there\;exists\;} a \in A \text{\;such\;that\;} e \leq a\}\).) Additionally, it is proven that when \(\sigma\) is the convex closure operator on a Euclidean space, \(\hat\sigma\) is the affine closure operator.