Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem

Abstract

In 1932, Pontrjagin and Schnirelmann [15] proved the classical theorem which characterizes topological dimension by use of box-counting dimensions. They proved their theorem by use of geometric arguments in some Euclidean spaces. In this paper, by use of dimensional theoretical techniques in an abstract topological space, we investigate strong relations between metrics of spaces and box-counting dimensions. First, by use of the numerical information of normal sequences of finite open covers of a space $X$, we prove directly the following theorem characterizing topological dimension dim $X$.

THEOREM 0.1. Let $X$ be a nonempty separable metric space. Then $$\mathrm{dim}X = \min \left\{ \displaystyle\liminf\limits_{i \to \infty} \frac{\log_3 | \: \mathcal{U}_i|}{i} \left.\right| \{\mathcal{U}\}^\infty_{i=1} \text{is a normal star-sequence of finite open covers of} X \text{and a development of} X \right\}$$ $$ = \min \left\{ \displaystyle\liminf\limits_{i \to \infty} \frac{\log_2 | \: \mathcal{U}_i|} {i} \left.\right| \{\mathcal{U}_i\}^\infty_{i = 1} \text{is a normal delta-sequence of finite open covers of} X \text{and a development of} X \right\}.$$ Next, we study box-counting dimensions $\dim_B(X,d)$ by use of Alexandroff-Urysohn metrics $d$ induced by normal sequences. We show that the above theorem implies Pontrjagin-Schnirelmann theorem. The proof is different from the one of Pontrjagin and Schnirelmann (see [15]). By use of normal sequences, we can construct freely metrics $d$ which control the values of $\log N (\epsilon,d) / | \log \epsilon|$. In particular, we can construct chaotic metrics with respect to the determination of the box-counting dimensions as follows.