## Convergence of holomorphic chains.(English)Zbl 0873.32005

A holomorphic $$p-$$chain in an open subset $$\varOmega$$ of $$\mathbb C^n$$ is a formal locally finite sum $$Z=\sum_{j\in J}k_jZ_j$$ where $$Z_j$$ are pairwise distinct irreducible analytic subsets of $$\varOmega$$ of pure dimension $$p$$ and $$k_j\in\mathbb Z\setminus\{0\}$$ for $$j\in J.$$ The set $$\mathcal G^p(\varOmega)$$ of holomorphic $$p-$$chains in $$\varOmega$$ is endowed with the structure of a free $$\mathbb Z-$$module.
The main aim of this note is to define a topology on $$\mathcal G^p(\varOmega)$$ and to study some properties of this topological space. The result of this construction is second-countable, metrizable, and convergence in it coincides with the one defined in E. M. Chirka [‘Complex analytic sets’, Kluwer Acad. Publishers (1989; Zbl 0683.32002)]. The topology constructed here is useful in studying the intersections of analytic sets [P. Tworzewski, Ann. Polon. Math. 62, No. 2, 177-191 (1995)].

### MSC:

 32B15 Analytic subsets of affine space 32C25 Analytic subsets and submanifolds 32C30 Integration on analytic sets and spaces, currents

### Keywords:

holomorphic chains; currents; convergence of chains

Zbl 0683.32002
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