Axiomatics of the dimension of metric spaces

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 $dA<dX$ and $X\setminus A$ is not connected.
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