---
tag: BX5WSK8ZCT
kind: concept
name: Semialgebraic set
msc: [14P10]
sources:
  - url: https://en.wikipedia.org/wiki/Semialgebraic_set
same_as:
  wikidata: Q7449354
  wikipedia: Semialgebraic set
---

## Summary

A semialgebraic set is a subset of $\R^n$ described by finitely many polynomial equations and strict inequalities combined with "and", "or" and "not".
By the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], the projection of a semialgebraic set is again semialgebraic.
Semialgebraic sets have finitely many connected components and admit triangulations.
They are the natural objects of [[real algebraic geometry|PPWJ8NFH7G]] and the model example of an [[o-minimal structure|DTWHAW1N0V]].

## Explanations

### L1 human

The closed disc $x^2 + y^2 \le 1$ is not the zero set of polynomials, but it is described by one polynomial inequality.
Semialgebraic sets are all shapes obtained in this way from finitely many polynomial equations and inequalities.
A half-plane, a triangle and the region between two parabolas are examples.
The graph of the sine function is not, because it crosses the axis infinitely often, while a polynomial description allows only finitely many such crossings.