real analytic subvariety

Let $U\subset {ℝ}^N$ be an open set.

Definition.

A closed set $X\subset U$ is called a real analytic subvarietyof $U$ such that for each point $p\in X$, there exists a neigbourhood$V$ and a set $ℱ$ of real analytic functions defined in $V$, such that

$$X\cap V=\{p\in V\mid f(p)=0\text{ for all }f\in ℱ\}.$$

If $U={ℝ}^N$ and all the $f\in ℱ$ are real polynomials, then$X$ is said to be a real algebraic subvariety.

If $X$ is not required to be closed, then it is said to be a local real analytic subvariety.Sometimes $X$ is called a real analytic set or real analytic variety. Similarly as for complexanalytic sets we can also define the regular and singular points.

Definition.

A point $p\in X$ is called a regular point if there is a neighbourhood$V$ of $p$ such that $X\cap V$ is a submanifold. Any otherpoint is called a singular point.

The set of regular points of $X$ is denoted by $X^{-}$ or sometimes $X^{*}.$ The set of singular pointsis no longer a subvariety as in the complex case, though it can be sown to be semianalytic. In general, real subvarieties is far worse behaved than their complex counterparts.