# Computability of semicomputable manifolds in computable topological spaces

## TL;DR

This paper investigates conditions under which semicomputable manifolds in computable topological spaces are actually computable, providing criteria involving the computability of their boundaries and compactification methods.

## Contribution

It establishes that a semicomputable compact manifold is computable if its boundary is computable, and extends this to certain noncompact manifolds via compactification techniques.

## Key findings

- Semicomputable compact manifolds with computable boundary are computable.
- Certain noncompact semicomputable manifolds become computable through compactification.
- Provides conditions linking boundary computability to overall manifold computability.

## Abstract

We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is computable if its boundary $\partial M$ is computable. We also show how this result combined with certain construction which compactifies a semicomputable set leads to the conclusion that some noncompact semicomputable manifolds in computable metric spaces are computable.