Contents

# Contents

## Idea

The Jordan curve theorem is a basic fact in topology. It stands out as having a statement that intuitively seems completely obvious, but which turns out to require a non-trivial formal proof.

The theorem states that every continuous loop (where a loop is a closed curve) in the Euclidean plane which does not intersect itself (a Jordan curve) divides the plane into two disjoint subsets (the connected components of the curve's complement), a bounded region inside the curve, and an unbounded region outside of it, each of which has the original curve as its boundary.

## Generalization

The Jordan–Brouwer separation theorem states that, in the Cartesian space $\mathbb{R}^n$, every simple closed continuous hypersurface (defined as the image of a continuous injection from the sphere $S^{n-1}$) has an exterior with two connected components, one bounded (the inside of the hypersurface) and one unbounded, each of which has the original hypersurface as its boundary. (Furthermore, the continuous map defining the hypersurface can be extended to an auto-homeomorphism of $\mathbb{R}^n$ under which the inside and outside appear as the images of the inside and outside of the sphere.)