---
tag: K1HTQF26MA
kind: concept
name: Subanalytic set
msc: [32B20, 14P15]
sources:
  - url: https://en.wikipedia.org/wiki/Subanalytic_set
same_as:
  wikidata: Q7630735
  wikipedia: Subanalytic set
---

## Summary

A subanalytic set is a subset of a [[real analytic manifold|Z7H98E1NHP]] that is locally the image of a bounded [[semianalytic set|E1V22QQHYJ]] under a coordinate [[projection|033DA1WE8Y]].
The class is larger than the class of semianalytic sets, because projections of semianalytic sets need not be semianalytic.
Subanalytic sets are closed under finite unions, intersections, complements and images under proper analytic maps.
They play in real analytic geometry the role that [[semialgebraic sets|BX5WSK8ZCT]] play in real algebraic geometry.

## Explanations

### L1 human

A semianalytic set is cut out locally by finitely many equations and inequalities between real analytic functions.
Shadows of such sets, obtained by forgetting some coordinates, can be more complicated.
Subanalytic sets are exactly these shadows, taken locally and of bounded pieces.
For example, the image of the bounded set $\set{(x, t) : x = t^2,\ 0 < t < 1}$ under $(x, t) \mapsto x$ is the interval $(0, 1)$, which is subanalytic.
In spite of the larger class, subanalytic sets still behave well: near each point they have only finitely many connected components.