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.