subanalytic set

Let $U\subset {ℝ}^n$.Suppose $𝒜(U)$ is any ring of real valued functions on$U$.Define $𝒮(𝒜(U))$ to be the smallestset of subsets of $U$, which contain the sets$\{x\in U\mid f(x)>0\}$ for all $f\in 𝒜(U)$,and is closed under finite union, finite intersection and complement.

Definition.

A set $V\subset {ℝ}^n$ is semianalyticif and only if for each $x\in {ℝ}^n$, there exists a neighbourhood$U$ of $x$, such that $V\cap U\in 𝒮(𝒪(U))$, where $𝒪(U)$denotes the real-analytic real valued functions.

Unlike for semialgebraic sets, there is no Tarski-Seidenberg theorem for semianalytic sets, and projections of semianalytic sets are in general not semianalytic.

Definition.

We say$V\subset {ℝ}^n$ is a subanalytic set if foreach $x\in {ℝ}^n$, there exists a relatively compact semianalytic set$X\subset {ℝ}^{n+m}$ and a neighbourhood $U$ of $x$, such that$V\cap U$ is the projection of $X$ onto the first $n$ coordinates.

In particular all semianalytic sets are subanalytic.On an open dense setsubanalytic sets are submanifolds and hence wecan define dimension. Hence at a point $p$, where a set $A$ is a submanifold,the dimension ${dim}_{p}A$ is the dimension of the submanifold. The dimension of the subanalytic set is the maximum ${dim}_{p}A$ for all$p$ where $A$ is a submanifold.Semianalytic sets are contained in a real-analytic subvariety of the same dimension. However, subanalytic sets are not in generalcontained in any subvariety of the same dimension. We do have however thefollowing.

Theorem.

A subanalytic set $A$ can be written as a locally finite union ofsubmanifolds.

The set of subanalytic sets is still not completely closed under projections however. Note thata real-analytic subvariety that is not relatively compact can have aprojection which is not a locally finite union of submanifolds, and henceis not subanalytic.

Definition.

Let $U\subset {ℝ}^n$. A mapping $f:U\to {ℝ}^m$ is said to be subanalytic (resp. semianalytic)if the graph of $f$ (i.e. the set $\{(x,y)\in U\times {ℝ}^m\mid x,y=f(x)\}$) is subanalytic (resp. semianalytic)