Topological entropy dimension for noncompact sets

Similar with the fractal dimension, we introduce the concept of topological entropy dimension to classify the sets with entropy zero. We prove that the entropy dimension of the space in this article is not greater than that defined by De Carvalho, where he introduced the entropy dimension for the system, and give some examples indicating that such inequality is optimal. Some basic propositions of entropy dimension are discussed and it turns out that the entropy dimension is invariant under conjugacy. The property of the countable stability and a power rule for the entropy dimension of any set are obtained. It is shown that any set shares the same entropy dimension with its image set.