Abstract

Let $X\subset {\mathbb{C}^{n}}^n$ be a closed real-analytic subset and put $$ \mathcal{A}:=\lbrace z\in X\mid \exists \ A\subset X,\text{ germ of a complex-analytic set, } z\in A,\, \dim _{z} A>0 \rbrace $$ This article deals with the question of the structure of $\mathcal{A}$. In the main result a natural proof is given for the fact, that $\mathcal{A}$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited