Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.15123 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909852230483968 |
|---|---|
| author | Bridges, Douglas S. |
| author_facet | Bridges, Douglas S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric Double Complements of Convex Sets Bridges, Douglas S. Logic 03F60, 52A05 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. |
| title | Metric Double Complements of Convex Sets |
| topic | Logic 03F60, 52A05 |
| url | https://arxiv.org/abs/2510.15123 |