Proposition 3

Let $X$ be a topological space.

a) For every nonempty closed subset $Y$ of $X$, one has

b) If $Y, Z, T$ are closed subsets of $X$ such that $Y \subset Z \subset T$, one has

It is enough to prove assertion a) in the case where $\dim(X)$ is finite. In this case, $\dim(Y)$ and $\operatorname{codim}(Y, X)$ are finite. There exists a chain $Y_0 \subset ... \subset Y_n$ of irreducible closed subsets of $Y$, of length $n = \dim(Y)$, and a chain $Y_n \subset ... \subset Y_{n+p}$ of irreducible closed subsets of $X$, of length $p \geq \operatorname{codim}(Y, X)$. It follows that $\dim(X) \geq n + p$, whence a). To establish b), one may suppose $Y$ irreducible. Since one has $\operatorname{codim}(Y, Z) \leq \operatorname{codim}(Y, T)$, the inequality is proved if $\operatorname{codim}(Y, Z) = +\infty$. Otherwise, let $Z_0$ be an irreducible component of $Z$ containing $Y$ and such that $\operatorname{codim}(Y, Z) = \operatorname{codim}(Y, Z_0)$. One has $\operatorname{codim}(Z, T) \leq \operatorname{codim}(Z_0, T)$, and one sees, as above, that $\operatorname{codim}(Y, Z_0) + \operatorname{codim}(Z_0, T) \leq \operatorname{codim}(Y, T)$, whence b).