Computability of 1-manifolds

TL;DR

This paper investigates the conditions under which semi-computable 1-manifolds with boundary are fully computable, especially focusing on non-compact cases with finitely many components.

Contribution

It extends previous results by demonstrating that semi-computable 1-manifolds with boundary and finitely many components are computable, even if not compact.

Findings

Semi-computable 1-manifolds with boundary and finitely many components are computable.

The result generalizes known theorems from compact manifolds to certain non-compact cases.

Finiteness of components is crucial for computability in this context.

Abstract

A semi-computable set S in a computable metric space need not be computable. However, in some cases, if S has certain topological properties, we can conclude that S is computable. It is known that if a semi-computable set S is a compact manifold with boundary, then the computability of \deltaS implies the computability of S. In this paper we examine the case when S is a 1-manifold with boundary, not necessarily compact. We show that a similar result holds in this case under assumption that S has finitely many components.