# Strukture izračunljivosti

Kulović, Edita (2016) Strukture izračunljivosti. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this paper, we expanded the notion of a recursive function to the function with real values. We have proven that the sum, the absolute value and the product of the recursive functions is also a recursive function. We introduced the notion of a recursively enumerable set. The preimage and image of a recursively enumerable set under the recursive function is a recursively enumerable set. In the third chapter we introduced the notions of a computable metric space, a computable point and a computable sequence. Let $d$ be the euclidean metric on $\mathbb{R}$ and $\alpha$ sequence introduced naturally such that $(\mathbb{R}, d, \alpha)$ is a computable metric space. We proved that x is a computable point in $(\mathbb{R}, d, \alpha)$ if and only if x is a recursive number. In the fourth chapter we defined the notion of a computability structure, a effective separating sequence, a separable computability structure etc. We have shown that the set of all recursive numbers is countable. The main result is a theorem in which we claim that there is a unique separable computability structure on the metric space $([0,1], d)$, where $d$ is the euclidean metric. Finally we examined boundedness in metric space and defined when a computable metric space is effectively totally bounded.