Real and complex analytic sets. The relevance of Segre varieties

Let $X\subset {{ℂ}^n}^n$ be a closed real-analytic subset and put

$$𝒜:=\{z\in X\mid \exists A\subset X,\text{germ}\text{of}\text{a}\text{complex-analytic}\text{set,}z\in A,{dim}_{z}A>0\}$$

This article deals with the question of the structure of $𝒜$. In the main result a natural proof is given for the fact, that $𝒜$ 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.