PomeCorollary 9.4.9.EE1633042D3D The normalization of a unibranch complex space is a homeomorphism. Every point $x$ of a complex space with $(X,x)$ irreducible, in particular every point of a unibranch space, has arbitrarily small neighborhoods $U$ such that $U\setminus A$ is connected for every nowhere dense analytic subset $A$ of $U$.
Corollary 9.4.9EE1633042D3D
Proof. On a unibranch space the normalization has one-point fibers, so it is a continuous bijection; being proper, it is closed, hence a homeomorphism. For the second assertion, let $U$ be the intersection of $X$ with a parametrization polydisc of $(X,x)$. Its covering locus is a connected manifold, dense in $U$; it remains connected after deleting $A$, by lemma 9.2.9, because $A$ is nowhere dense, and the result is dense in $U\setminus A$. Hence $U\setminus A$ is connected. $\square$
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