Corollary 9.3.1727E8771DE67A

Proof. By the uniqueness in corollary 9.3.16, applied to the family consisting of $Z$ alone, $Z$ is its own only irreducible component; so $Z_{\mathrm{reg}}$ is connected and dense in $Z$, and its dimension is constant. (1) Let $B$ be a branch of $(Z,x)$. By corollary 9.3.6, $B$ contains points of the covering locus of a parametrization of $B$ outside the other branches of $(Z,x)$. Near such a point, $Z$ coincides with $B$ and is a manifold of dimension $\dim B$, so the point lies in $Z_{\mathrm{reg}}$, and $\dim B=\dim Z_{\mathrm{reg}}$.

(2) The analytic subset $Y\cap Z_{\mathrm{reg}}$ of the connected manifold $Z_{\mathrm{reg}}$ has empty interior in $Z_{\mathrm{reg}}$: otherwise it would be all of $Z_{\mathrm{reg}}$, by lemma 9.2.8, and the closed set $Y$ would contain the closure $Z$ of $Z_{\mathrm{reg}}$. Since $Z_{\mathrm{reg}}$ is open and dense in $Z$, it follows that the closed set $Y$ has empty interior in $Z$. Now let $x\in Y$. Every branch $Y_1$ of $(Y,x)$ is contained in a branch $B$ of $(Z,x)$, and it differs from $B$: otherwise $Y$ would contain the points of $B$ near $x$ outside the other branches of $(Z,x)$, which form a nonempty open subset of $Z$ by corollary 9.3.6. By proposition 9.3.9 and (1), $\dim Y_1<\dim B=\dim Z$.

(3) Let $e=\dim Z=\dim Z'$. By corollary 9.3.13(1) and by (1), $Z'_{\mathrm{sing}}$ has dimension less than $e$ at each of its points, so by (2), applied to $Z'_{\mathrm{sing}}\cap Z\subset Z$, there is a point $p\in Z_{\mathrm{reg}}\setminus Z'_{\mathrm{sing}}$. Near $p$, the set $Z$ is a submanifold of dimension $e$ of the manifold $Z'$ of dimension $e$; since the germ of a manifold is irreducible (lemma 9.3.12) and proper subgerms of irreducible germs have smaller dimension (proposition 9.3.9), $Z$ and $Z'$ coincide near $p$. Thus the analytic subset $Z$ of $Z'$ has interior points in $Z'$, and $Z=Z'$ by (2).

(4) Let $Z\subset \bigcup _\beta Y_\beta $ with $(Y_\beta )$ locally finite. The sets $Z_{\mathrm{reg}}\cap Y_\beta $ are closed analytic subsets of the connected manifold $Z_{\mathrm{reg}}$ with union $Z_{\mathrm{reg}}$. A locally finite union of closed nowhere dense sets is closed and nowhere dense, so one of them has interior points and is therefore all of $Z_{\mathrm{reg}}$. Then $Z\subset Y_\beta $, because $Y_\beta $ is closed and $Z_{\mathrm{reg}}$ is dense in $Z$. $\square$