Homeomorphism between Sphere and Cube

In this paper, we present the construction of homeomorphism from unit sphere; S 2 to unit cube; C 3 . On route, we produced an explicit mapping between the two topological spaces where proving by construction is mainly adopted in this work. | Homeomorphism | Homeomorphic | Continuously deformed |


Introduction
Sphere and cube are two different geometrical objects. Even though a sphere and a cube have a different shape, topologically there is no difference between them [1]. In that case, we can say that the surface of a sphere is a topologically equivalent to the surface of a cube. Two structures are topologically equivalent if and only if one shape can be continuously deformed to the other shape such as bending, stretching or squeezing without being severed, tearing or gluing [2]. In other words, there exists homeomorphism that is an open continuous bijection mapping between sphere and cube.
We consider two surfaces in space: a sphere and a cube. We can see that the sphere can be continuously deformed into the cube without tearing or collapsing them [3]. Many literature [3], [4], [5], [6], [7] mentioned that it is not hard to convince that a sphere can be deformed into a cube. However not all of them come up with an explicit mapping for a homeomorphism between them. Even though J.M. Lee in [3] posted the mapping from a sphere to a cube, he left out most of the essentials proofs to be homeomorphics.

C 3 as a topological space
Before we prove that C 3 is a topological space, we defined the cube C 3 as follows. We denote C 3 as a surface of the unit cube [ ] ( ) 3 3 1,1 \ 1,1 − − bounded by six square faces centered at the origin in three-dimension ( Figure 1).
Next, we show that C 3 is a topological space as follows.
We define the open set V on First, we need to show that ∅ ∈ 3 C τ and C 3 ∈ 3 C τ . We know that ∅ is a subset of any set [9] and it is open [10]. Therefore ∅ is an open set for C 3 . Thus, ∅ ∈ 3 C τ . We also know that R 3 is open and C 3 is a subset of R 3 .
Next, we show that Lastly, we show that a topology on C 3 and thus (C 3 , 3 C τ ) is a topological space.
In the next section, the construction of mapping between unit sphere, S 2 to unit cube, C 3 will be given.

Mapping from S 2 to C 3
In this section, we define the mapping from S 2 to C 3 as follows: , y≥ x, y>z} , On the other hand, we partition the unit cube as So, the mapping can be redefined as : Then, we build the following lemmas in order to prove that S 2 is homeomorphic to C 3 . Therefore max{|x|} = max{|y|} = a and min{|z|} = a .

Lemma 3.2 If
, then Ø is onto.

Lemma 3.12 If
In this section, we have shown that Ø is a function from

Homeomorphism between S 2 and C 3
In this section, we will prove the equivalent structure of the sphere and cube (i.e. homeomorphism). In other words, S 2 ≅ C 3 . Before that, we present some general definitions and theorems which will be used along with the construction of this homeomorphism.

Theorem 4.1 [11]
If F = (f 1 , f 2 , …, f m ) is a mapping from R n to R m , then

Definition 4.4 [12]
A function f : X →Y between topological space is called a homeomorphism if f : X →Y is one-to-one and onto and both f and f -1 are continuous. The notation X ≅ Y means that X is homeomorphic to Y. , , x y z . Therefore , , , , x y z . Thus, Ø is a function from a topological space S 2 to a topological space C 3 . In the following proving, we will consider the function as Ø : Next, we will show Ø is bijective. Pick a∈ C 3 . Therefore ∃ (x, y, z) ∈ S 2 ∋ a = ( , , ) max{ , , } x y z x y z = Ø(x, y, z). Thus Ø is onto. Now, we will use the same proving technique that had been done by Tahir [8], [13 ] to show that Ø is one to one. Let Ø = Ø* and From (3), we have From the components of Jacobian matrix above, we can conclude that This implies Thus, Ø is one-to-one. Since Ø is onto and one-to-one, therefore Ø is a bijection.
Next, we will show that Ø is continuous by using Theorem 4.2. According to this theorem, we need to show that Ø -1 (β) is open in S 2 for every open set β in C 3 . Pick (a, b, c Since Ø -1 (β) is a bijection, there exists unique (x', y', z') ∈ β such that Ø -1 (x', y', z') = (a, b, c).

Figure 4 :
Homeomorphism of S 2 and C 3

Conclusion
In this paper, we have shown the construction of the homeomorphism between the unit sphere and the unit cube where they have an equivalent structure. It also preserves properties that spaces have. Results obtained from this research are very important in image processing and particularly Electroencephalography (EEG) signal for human brain.