Computational Dimension of Topological Spaces

Abstract

When a topological space X can be embedded into the space Σ ^w_⊥,n , of n⊥-sequences of Σ, then we can define the corresponding computational notion over X because a machine with n+1 heads on each tape can input/output sequences inΣ ^w_⊥,n ⋅. This means that the least number n such that X can be topologically embedded into Σ ^w_{⊥, n} serves as a degree of complexity of the space. We prove that this number, which we call the computational dimension of the space, is equal to the topological dimension for separable metric spaces. First, we show that the weak inductive dimension of Σ ^w_⊥,n is n, and thus the computational dimension is at least as large as the weak inductive dimension for all spaces. Then, we show that the Nöbeling’s universal n-dimensional space can be embedded into Σ ^w_{⊥, n} and thus the computational dimension is at most as large as the weak inductive dimension for separable metric spaces. As a corollary, the 2-dimensional Euclidean space ℝ² can be embedded in {0,1} ^w_⊥,2 but not in Σ ^w_⊥,1 for any character set Σ, and infinite dimensional spaces like the set of closed/open/compact subsets of IR ^m and the set of continuous functions from ℝ^l to ℝ^m can be embedded in Σ ^w_⋅ but not in : ^w_⊥,n for any n.

Acknowledgement

The author thanks Andreas Knobel and Izumi Takeuti for many illuminating discussions, and anonymous referees for invaluable comments.