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Continuity of intersection of analytic sets. (English) Zbl 0576.32013
Let X be a metric space; let \({\mathcal F}_ X\) be the family of all closed subsets of X. We say that, given \((F_ n)_{n\in N}\subset {\mathcal F}_ X,\) \(F_ n\to F\in {\mathcal F}_ X\) if (a) for every \(x\in F\) and every neighbourhood V of x, \(F_ n\cap V=\emptyset\) for at most finitely many n, and (b) for every \(x\not\in F\), there exists a neighbourhood U of x such that \(F_ n\cap U\neq \emptyset\) for at most finitely many n.
Using this notion of convergence, which is the natural extension of Hausdorff convergence to the noncompact case, the authors prove the following continuity theorem for intersections of complex analytic subsets.
Theorem: Let \(\Omega\) be an open subset of \({\mathbb{C}}^ n\) and let \({\mathcal A}_ p(\Omega)\) be the set of pure p-dimensional complex analytic subsets of \(\Omega\) ; let \(V_ 0\in {\mathcal A}_ p(\Omega)\), \(W_ 0\in {\mathcal A}_ q(\Omega)\) such that \(p+q\geq n\) and \(V_ 0\cap W_ 0\in {\mathcal A}(\Omega)_{p+q-n};\) then the mapping \(\cap: {\mathcal A}_ p(\Omega)\times {\mathcal A}_ q(\Omega)\to {\mathcal F}_{\Omega},\) \((V,W)\mapsto V\cap W\) is continuous at the point \((V_ 0,W_ 0)\) with respect to the described topology.

MSC:
32B15 Analytic subsets of affine space
32C25 Analytic subsets and submanifolds
32A05 Power series, series of functions of several complex variables
32A10 Holomorphic functions of several complex variables
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