nLab
Jordan curve theorem
Contents
Context
Topology
topology (point-set topology , point-free topology )

see also differential topology , algebraic topology , functional analysis and topological homotopy theory

Introduction

Basic concepts

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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.)

References
Last revised on May 6, 2021 at 01:10:23.
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