关于整体定义的半解析集
On globally defined semianalytic sets
摘要：
在这项工作中，我们提出的概念$C$ -半解析子集的一个真正的解析流形和更一般的一个真正的解析空间。$C$ -半解析集可以理解为对Cartan（$C$ -解析集）引入的整体解析集的半解析集的自然推广。更确切地说，$S$是一个$C$ -半解析子集的一个真正的分析空间$(X,{\mathcal O}_X)$，如果每个点的$X$有一个邻域$U$，使$S\cap U$是一个有限的布尔组合的全球分析等式和严格不等式的$X$。利用仿紧性，$C$ -半解析集是$X$上的整体解析等式和严格不等式的有限布尔组合的局部有限并. $C$ -半解析集族在与半解析集族相同的运算下是封闭的：局部有限并和交、补、闭包、内部、连通分量、解析映射下的逆像、维数为$k$的点集等，尽管它们的定义只涉及全局解析函数。此外，我们还将次解析集刻画为$C$ -半解析集在真解析映射下的象.我们还证明了$C$ -半解析集$S$在Stein空间之间的真全纯映射下的象也是$C$ -半解析集。前面的结果使我们能够更好地理解实解析流形$M$的$C$ -解析子集$X$的非相干点集$N(X)$的结构.我们提供了一个全球性的几何拓扑描述$N(X)$的启发，由于Tanqui-Tognoli（1980），这需要复杂的解析规范化的解析集相应的本地。因此，它认为$N(X)$是一个$C$ -维数为$\leq\dim(X)-2$的半解析集.
英文摘要：
In this work we present the concept of $C$-semianalytic subset of a real analytic manifold and more generally of a real analytic space. $C$-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan ($C$-analytic sets for short). More precisely $S$ is a $C$-semianalytic subset of a real analytic space $(X,{\mathcal O}_X)$ if each point of $X$ has a neighborhood $U$ such that $S\cap U$ is a finite boolean combinations of global analytic equalities and strict inequalities on $X$. By means of paracompactness $C$-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on $X$. The family of $C$-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension $k$, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of $C$-semianalytic sets. We prove also that the image of a $C$-semianalytic set $S$ under a proper holomorphic map between Stein spaces is again a $C$-semianalytic set. The previous result allows us to understand better the structure of the set $N(X)$ of points of non-coherence of a $C$-analytic subset $X$ of a real analytic manifold $M$. We provide a global geometric-topological description of $N(X)$ inspired by the corresponding local one for analytic sets due to Tancredi-Tognoli (1980), which requires complex analytic normalization. As a consequence it holds that $N(X)$ is a $C$-semianalytic set of dimension $\leq\dim(X)-2$.
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