2025/10/16 by Douglas Bridges, Bridges, Douglas S.
Mathematics · #03F60 #52A05 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2510.15123
openalex publication_date 2025/10/16 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28
In constructive mathematics the metric complement of a subset S of a metric space X is the set -S of points in X that are bounded away from S. In this note we discuss, within Bishop's constructive mathematics, the connection between the metric double complement, -(-K), and the logical double complement, not not K, where K is a convex subset of a normed linear space X. In particular, we prove that if K has inhabited interior, then -(-K) equals the interior of not not K, that the hypothesis of inhabited interior can be dropped in the finite-dimensional case, and that we cannot constructively replace the interior of not not K by that of K in these results.