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Using the local flattening theorem we prove that the set of points where a complex analytic morphism is open (or universally open) is constructible. A similar result is obtained for a morphism of complex algebraic varieties. Let f: X -* Y be a morphism of schemes and let L(f ) C X be the set of points where f is universally open (see Definition 1.2). In [EGA, (14.3.9)] Grothendieck asked whether L(f ) is always a constructible subset of X. If X and Y are of finite type over a field k and f is a morphism over k, then the answer is positive and was given by Roland Huber [Hu], who also posed the question whether the same is true in the complex analytic category. We shall show in ? 1 of this paper that this is the case (Theorem 1.4). The idea of our proof is different from that of Huber. His proof is based on the properties of real closure of a field and real spectrum (or valuation spectrum in the case of a field of positive characteristic) and ours uses flatness, namely, the local flattening theorem [HLT, Theoreme 4; Hi, Theorem 4.4]. In ?2 we show how to extend our proof to the category of complex algebraic varieties. At the end we give two interesting examples of L(f ) . In view of Hubert's proof it is interesting to ask a similar question in the real analytic category. Question. Let f: X -* Y be a morphism of real analytic spaces. Is L(f ) always a semi-analytic subset of X ? As Huber pointed out to me, it is easy to see that in this case L(f ) is a subanalytic subset of X. 1. COMPLEX ANALYTIC CASE Definition 1.1. Let f: V -* W be a continuous map of topological spaces. We say that f is open at x E V if the image of each neighbourhood of x is a neighbourhood of f(x). Note that the above condition does not mean that the image of any open neighbourhood of x is an open (!) neighbourhood of f(x). If f is open at Received by the editors May 28, 1991. 1991 Mathematics Subject Classification. Primary 32C1 5, 14E40.

Constructibility of the set of points where a complex analytic morphism is open