5. Computability on Closed, Open and Compact Sets

Abstract

Since the set 2^ℝ of all subsets of ℝ has a cardinality greater than that of Σ^ω, it has no representation. Therefore, the framework of TTE cannot be applied to define computable functions on all subsets of ℝ like sup :⊆ 2^ℝ → ℝ, which determines the least upper bound for each set X ⊆ ℝ with upper bound. In this section we select three subclasses of subsets of ℝⁿ which have continuum cardinality, the open, the closed and the compact sets. As llsual for a subset X of ℝ, we denote the closure of X by X̄ and the open kernel or interior of X by X^o. We will use the notation Iⁿ of the set Cb⁽ⁿ⁾ of open rational cubes (balls) in ℝⁿ from Definition 4.1.2. By Īⁿ(ω) we denote the closure of the open cube Iⁿ(ω).