# Computability of 1-manifolds

Burnik, Konrad (2015) Computability of 1-manifolds. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this paper we investigate whether some analogue of the implication that was proved in [9] for semi-computable compact manifolds M and which states $\partial M$ computably compact $\Rightarrow M$ computably compact $(\star)$ also holds when $M$ is a 1-manifold in some computable metric space. In the first introductory chapter we give an overview of the known results regarding the implication $(\star)$. In chapter 2 we give an overview of the basic notions from classic computability theory which we use in subsequent chapters. In chapter 3 we generalize computability to metric spaces. A computable metric space is the main ambient space in which we state and prove our main results. Chapter 4 is preparatory. In this chapter, we introduce the notions of formal computability properties occurring in computable metric spaces. The effective covering property, together with the property of compact closed balls is shown to be key if we want to prove that each co-recursively enumerable topological ray with a computable endpoint is computable. A sufficient condition when a metric space has the effective covering property [6] is stated and proved in chapter 5. At the end of that chapter, we write out a proof that $\mathbb{R}^n$ has the effective covering property. It turns out that the effective covering property is not always necessary to prove the main implication $(\star)$. Therefore, in chapter 6 we introduce new notions of semi-computable compact on closed balls and computable compact on closed balls. In the case when the ambient space does have the effective covering property and compact closed balls, then the notion of semi-computable compact on closed balls is equivalent to the notion of corecursively enumerable set. The effective covering property and the property of having compact closed balls are properties that hold uniformly in the whole space, which are strong assumptions, so to obtain more general results we had replaced them with a local property of semi-computable compactness on closed balls. Therefore, the main results for the topological ray and line we state and prove without the assumption on the ambient space that it has the effective covering property and compact closed balls and instead we use the newly introduced notions. In chapter 7 we give the proof of the main result for the topological ray and in chapter 8 we prove a similar result for the topological line. In chapter 9 we study further the necessity of additional conditions on the ambient space. We give examples of spaces which show that in case the computable metric space has exactly one of the two additional conditions then the conclusions of chapters 7 and 8 fail to be true. Together with the well known results on computability of chainable and circularly chainable continua given in [6], the results for the topological ray and line have led us to the main result of this thesis, the computability of 1-manifolds. In the final chapter 10 we study the computability of 1-manifolds in computable metric spaces and prove the main result of this thesis: every semicomputable compact on closed balls 1-manifold with computable boundary and finitely many connected components is computable compact on closed balls.