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The proof of the Jordan Curve Theorem (JCT) in this paper is focused on a graphic illustration and analysis ways so as to make the topological proof more understandable, and is based on the Tverberg’s method, which is acknowledged as being quite esoteric with no graphic explanations. The preliminary constructs a parametrisation model for Jordan Polygons. It takes quite a length to introduce four lemmas since the proof by Jordan Polygon is the approach we want to concern about. Lemmas show that JCT holds for Jordan polygon and Jordan curve could be approximated uniformly by a sequence of Jordan polygons. Also, lemmas provide a certain metric description of Jordan polygons to help evaluate the limit. The final part is the proof of the theorem on the premise of introduced preliminary and lemmas.

Though the definition of the Jordan Curve Theorem is not hermetic at all, the proof of the theorem is quite formidable and has experienced ups and downs throughout history.

Bernard Bolzano was the first person who formulated a precise conjecture: it was not self-evident but required a hard proof. However, the Jordan Curve Theorem was named after Camille Jordan, a mathematician who came up with the first proof in his lectures on real analysis and published his findings in his book [

This paper is intended to strengthen Tverberg’s claim. Tverberg once suggested in his paper that “Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it. The present paper is intended to provide a reasonably short and self-contained proof or at least, failing that, to point at the need for one” [

It is comparatively easy to prove that the Jordan curve theorem holds for every Jordan polygon in Lemma 1, and every Jordan curve can be approximated arbitrarily well by a Jordan polygon in Lemma 2. Then Lemma 3 and Lemma 4 deal with the situation in limiting processes to prevent the cases from the polygons that may thin to zero somewhere.

Let C be the unit circle { ( x , y ) | x 2 + y 2 = 1 } , a Jordan curve Γ is the image of C under an injective continuous mapping γ into ℝ 2 , i.e., a simple closed curve on the plane. We have the following fundamental fact.

Jordan Curve Theorem [

We introduce here some terms from analysis that will be used later.

First, we may recall some basic knowledge in real analysis.

We say that M ∈ ℝ is a lower bound for a set S ⊂ ℝ if each s ∈ S satisfies s ≥ M . And a lower bound for S that is greater than all other lower bounds for S is a greatest lower bound for S. The greatest lower bound for S is denoted g . l . b ( S ) or i n f ( S ) .

Since C is compact, the mapping γ is uniformly continuous on C, its inverse γ − 1 : Γ → C is also uniformly continuous. Next, if A and B are nonempty disjoint compact sets in ℝ 2 , then

d ( A , B ) : = inf { | a − b | : a ∈ A , b ∈ B } [

Note that d ( A , B ) is always positive, since, otherwise we have a sequence of points ( a n ) , ( b n ) , n ∈ ℕ , so that | a i − b i | → 0 as i → ∞ . Since A is compact, ( a n ) has a subsequential limit, say a ∈ A , so that for any ϵ and sufficiently large N, by triangular inequality,

| a − b N | ≤ | a − a N | + | a N − b N | ≤ ϵ

then a is a limit point of B thus a ∈ B contradicts to A ∩ B = ∅ .

We assume that the unit circle C is consistent with the natural parametrisation t ( θ ) = ( cos θ , sin θ ) , θ ∈ [ 0 , 2 π ] . A Jordan curve Γ is said to be a Jordan polygon if there is a partition Θ = { θ 0 , θ 1 , ⋯ , θ n } of the interval [ 0 , 2 π ] (i.e., 0 = θ 0 < θ 1 < ⋯ < θ n = 2 π ), such that

γ ( ( cos θ , sin θ ) ) = ( a i θ + b i , c i θ + d i )

for θ ∈ [ θ i − 1 , θ i ] , i ∈ { 1, ⋯ , n } , a i , b i , c i , d i are all real constants.

We call the pair ( γ , Θ ) a realisation of a polygon Γ .

Remark. In accordance with the choice of partition Θ , we could define edges and vertices in a natural way. It is possible that adjacent edges of a polygon Γ lie on the same line (

We divide the proof of JCT into several steps. Lemma 1 below shows that JCT indeed holds for Jordan polygons. Lemma 2 shows every Jordan curve could be approximated uniformly by a sequence of Jordan polygons. Lemmas 3 and 4 provide certain metric description of Jordan polygons, which helps to evaluate the limit.

Lemma 1. The Jordan curve theorem holds for every Jordan polygon Γ with realisation ( γ , Θ ) .

Proof. Denote edges of Γ to be E 1 , E 2 , ⋯ , E n , and vertices to be v 1 , v 2 , ⋯ , v n , (so E i = γ ( ( θ i − 1 , θ i ) ) and v i = γ ( θ i ) ) with

E i ∩ E i + 1 = { v i } ( i = 1 , ⋯ , n , E n + 1 = E 1 , v i + 1 = v 1 ) [

1) ℝ 2 \ Γ has at most two components.

Let δ : = min { d ( E i , E j ) | E i and E j are not adjacent } and let N i : = { q ∈ ℝ 2 | d ( q , E i ) < δ } (

By definition no points of N i belongs to E j , where j ≠ i − 1, i , i + 1 (recall that E 0 = E n ). so N i ∩ Γ ⊂ E i − 1 ∪ E i ∪ E i + 1 and it is clear that N i \ Γ consists of two connected components, say, N ′ i and N ″ i . We may assume, by elementary analytic geometry, (

N ′ i ∩ N ′ i + 1 ≠ ∅ , N ″ i ∩ N ″ i + 1 ≠ ∅ , i = 1 , ⋯ , n

Therefore

Therefore

2)

We place

We say a line

We now partition

For every point

vertex

None of edges

For every

1)

2)

3)

4)

For case (1) we take

We have shown that every point sufficiently close to an odd (resp.even) point is odd (resp.even). If

then

It remains to show that there do exist even points and odd points.

Take any p so that

Take a sufficiently large open ball

We have shown that

Remark. The above proof has shown that a Jordan polygon divides the plane into a bounded connected component O and an unbounded connected component V. Moreover, O and V are nonempty and open.

Next we show that any Jordan curve

Lemma 2. Every Jordan Curve

Proof. We want to show that given any

By uniform continuity we can choose

and

Put

If we fill the plane by squares with vertices

Each

We now construct

We first change

on C containing

that when

The procedure would end in at most n steps as

By above argument each circular arc

Because

Remark. Notice the two points on the unit circle C of distance

an arc of radian

polygon one could produce on the unit circle. it helps us, in the above process, really produce a polygon for each

We can now approximate

a)

b)

The following lemmas 3 and 4 ensure that (a) and (b) mentioned above would not happen in the limiting process.

For every point

Lemma 3. Let

Proof. The existence of the disc D with

Let

Suppose

Now a and b could either belong to those vertices or not. We shall divide the situation into different cases (

1)

In this case the boundary of D is tangent to both

2)

The boundary of D passes

3)

We consider a variable circle through

meet

become tangent to segment

Consider

Lemma 4. Under the assumptions stated there is a continuous path

Proof. We first note that if

Hence we may assume now that

Choose

D arrives at

Suppose not. At some position, D and

same component, say Y, replacing

Now draw a unit circle E centred at b. The circle meets

As E lies in X, and

c arrives at b

We have verified that D reaches

The statement is clear when

The existence of unbounded component is clear. For the bounded one, draw a large closed disc D with boundary circle

By uniform continuity there exists

so z and

If z is in the unbounded component of

so for such n,

and some

Taking

For any

Assume first that there is some

continuous path

connected by a path not intersecting

By above argument we conclude that for any

1) p and q are in different components of

2)

Let T be the component of

In particular, we have

So far the proof of Jordan Curve Theorem is done. The four lemmas are of paramount importance in the whole proof. Especially, the first lemma, which Jordan was in an inappropriate way of demonstration, seems to be intuitive, however, requires rigorous proof. This theorem is discussed in the

I’m very grateful that the Professor Charles C. Pugh from the Department of Mathematics, University of California of Berkeley helped me a lot with my study of Topology. And I appreciate my groupmate Haiqi Wu from Oxford helped me with my paper. And thanks to the platform provided by CIS organization, I have the chance to study topology with the emeritus professor and hence accomplish the paper. Also if it were not for the teachers from my alma mater Wuhan University, I wouldn’t have been able to walk on my avenue of Mathematics with determination. Last but not least, I would like to thank my parents, and it’s they who are always supporting me behind.

The author declares no conflicts of interest regarding the publication of this paper.