subanalytic set
Let $U\u0e42\x8a\x82{\mathrm{\u0e42\x84\x9d}}^{n}$. Suppose $\mathrm{\u0e50\x9d\x92\x9c}\u0e42\x81\u0e02(U)$ is any ring of real valued functions on $U$. Define $\mathrm{\u0e50\x9d\x92\u0e0e}\u0e42\x81\u0e02(\mathrm{\u0e50\x9d\x92\x9c}\u0e42\x81\u0e02(U))$ to be the smallest set of subsets of $U$, which contain the sets $\{x\u0e42\x88\x88U\u0e42\x81\u0e02\u0e42\x88\u0e03f\u0e42\x81\u0e02(x)>\u0e42\x81\u0e020\}$ for all $f\u0e42\x88\x88\mathrm{\u0e50\x9d\x92\x9c}\u0e42\x81\u0e02(U)$, and is closed under finite union, finite intersection^{} and complement.
Definition.
A set $V\u0e42\x8a\x82{\mathrm{\u0e42\x84\x9d}}^{n}$ is semianalytic if and only if for each $x\u0e42\x88\x88{\mathrm{\u0e42\x84\x9d}}^{n}$, there exists a neighbourhood $U$ of $x$, such that $V\u0e42\x88\u0e09U\u0e42\x88\x88\mathrm{\u0e50\x9d\x92\u0e0e}\u0e42\x81\u0e02(\mathrm{\u0e50\x9d\x92\u0e0a}\u0e42\x81\u0e02(U))$, where $\mathrm{\u0e50\x9d\x92\u0e0a}\u0e42\x81\u0e02(U)$ 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 $V\u0e42\x8a\x82{\mathrm{\u0e42\x84\x9d}}^{n}$ is a subanalytic set if for each $x\u0e42\x88\x88{\mathrm{\u0e42\x84\x9d}}^{n}$, there exists a relatively compact semianalytic set $X\u0e42\x8a\x82{\mathrm{\u0e42\x84\x9d}}^{n+m}$ and a neighbourhood $U$ of $x$, such that $V\u0e42\x88\u0e09U$ is the projection of $X$ onto the first $n$ coordinates^{}.
In particular all semianalytic sets are subanalytic. On an open dense set subanalytic sets are submanifolds and hence we can define dimension^{}. Hence at a point $p$, where a set $A$ is a submanifold, the dimension ${dim}_{p}\u0e42\x81\u0e01A$ is the dimension of the submanifold. The dimension of the subanalytic set is the maximum ${dim}_{p}\u0e42\x81\u0e01A$ 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 general contained in any subvariety of the same dimension. We do have however the following.
Theorem.
A subanalytic set $A$ can be written as a locally finite^{} union of submanifolds.
The set of subanalytic sets is still not completely closed under projections however. Note that a real-analytic subvariety that is not relatively compact can have a projection which is not a locally finite union of submanifolds, and hence is not subanalytic.
Definition.
Let $U\u0e42\x8a\x82{\mathrm{\u0e42\x84\x9d}}^{n}$. A mapping $f:U\u0e42\x86\x92{\mathrm{\u0e42\x84\x9d}}^{m}$ is said to be subanalytic (resp. semianalytic) if the graph of $f$ (i.e. the set $\{(x,y)\u0e42\x88\x88U\u0e23\x97{\mathrm{\u0e42\x84\x9d}}^{m}\u0e42\x88\u0e03x,y=f\u0e42\x81\u0e02(x)\}$) is subanalytic (resp. semianalytic)
References
- 1 Edward Bierstone and Pierreย D. Milman, Semianalytic and subanalytic sets, Inst. Hautes รtudes Sci. Publ. Math. (1988), no.ย 67, 5โ42. http://www.ams.org/mathscinet-getitem?mr=89k:32011MR 89k:32011
Title | subanalytic set |
Canonical name | SubanalyticSet |
Date of creation | 2013-03-22 16:46:16 |
Last modified on | 2013-03-22 16:46:16 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 6 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 32B20 |
Classification | msc 14P15 |
Related topic | TarskiSeidenbergTheorem |
Related topic | SemialgebraicSet |
Defines | subanalytic |
Defines | semianalytic set |
Defines | semianalytic |
Defines | semianalytic function |
Defines | subanalytic function |
Defines | semianalytic mapping |
Defines | subanalytic mapping |
Defines | dimension of a subanalytic set |