Axiomatics of the dimension of metric spaces

E. V. Shchepin

Sbornik. Mathematics, Tome 21 (1973) no. 1, pp. 137-143

Résumé

In this paper we prove that there exists a unique function $\dim X$ which assigns to every finite-dimensional metric space $X$ an integer $dX$ such that the following axioms are satisfied. Axiom 1. $dT^n=n$ $(T^n$ is an $n$-dimensional simplex). \smallskip Axiom 2. $d\bigcup^\infty_iX_i=\max_idX_i$ if all $X_i$ are closed in $\bigcup^\infty_iX_i=X$. \smallskip Axiom 3. For every $X$ there exists a finite open cover $\omega$ such that $dY\geqslant dX$ for every $\omega$-mapping $f\colon X\to Y$. \smallskip Axiom 4. For every $X$ there exists a closed subset $A$ such that $dA$ and $X\setminus A$ is not connected. Bibliography: 2 titles.