String-Based Borsuk-Ulam Theorem and Wired Friend Theorem

# String-Based Borsuk-Ulam Theorem and Wired Friend Theorem

J.F. Peters Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Canada and Department of Mathematics, Faculty of Arts and Sciences, Adi̇yaman University, 02040 Adi̇yaman, Turkey  and  A. Tozzi Department of Physics, Center for Nonlinear Science, University of North Texas, Denton, Texas 76203 and Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Canada Dedicated to Som Naimpally, Mathematician and D.I. Olive, F.R.S.
###### Abstract.

This paper introduces a string-based extension of the Borsuk-Ulam Theorem (denoted by strBUT). A string is a region with zero width and either bounded or unbounded length on the surface of an -sphere or a region of a normed linear space. In this work, an -sphere surface is covered by a collection of strings. For a strongly proximal continuous function on an -sphere into -dimensional Euclidean space, there exists a pair of antipodal -sphere strings with matching descriptions that map into Euclidean space . Each region of a string-covered -sphere is a worldsheet. For a strongly proximal continuous mapping from a worldsheet-covered -sphere to , strongly near antipodal worldsheets map into the same region in . This leads to a wired friend theorem in descriptive string theory. An application of strBUT is given in terms of the evaluation of Electroencephalography (EEG) patterns.

###### Key words and phrases:
Borsuk-Ulam Theorem, -sphere, Region, Strong Proximity, String
###### 2010 Mathematics Subject Classification:
Primary 54E05 (Proximity structures); Secondary 37J05 (General Topology), 55P50 (String Topology)
The research has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211402463, Natural Sciences & Engineering Research Council of Canada (NSERC) discovery grant 185986 and Instituto Nazionale di Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036.

## 1. Introduction

In this paper, we consider a geometric structure that has the characteristics of a cosmological string (denoted ), which is the path followed by a particle moving through space.

A string is a region of space with zero width (in an abstract geometric space or geometry in a science of spaces [31, §8.0], esp. [6]) or non-zero width (in a non-abstract, physical geometry space such as the space described by V.F. Lenzen [14]) and either bounded or unbounded length. Another name for such a string is worldline [20, 19, 18]. Here, a string is a region on the surface of an -sphere or in an -dimensional normed linear space. Every string is a spatial region, but not every spatial region is a string. Strings that have some or no points in common are antipodal.

Examples of strings with finite length are shown in Fig. 1. Regions are examples of antipodal strings. Various continuous mappings from an -dimensional hypersphere to a feature space that is an -dimensional Euclidean space lead to a string-based incarnation of the Borsuk-Ulam Theorem, which is a remarkable finding about continuous mappings from antipodal points on an -sphere into , found by K. Borsuk [1].

The Borsuk-Ulam Theorem is given in the following form by M.C. Crabb and J. Jaworowski [8].

###### Theorem 1.

Borsuk-Ulam Theorem [1, Satz II].
Let be a continuous map. There exists a point such that .

###### Proof.

A proof of this form of the Borsuk-Ulam Theorem follows from a result given in 1930 by L.A. Lusternik and L. Schnirelmann [15]. For the backbone of the proof, see Remark 3.4 in [8]. ∎

## 2. Preliminaries

This section briefly introduces Petty antipodal sets that are strings and the usual continuous function required by BUT is replaced by a proximally continuous function. In addition, we briefly consider strong descriptive proximities as well as two point-based variations of the Borsuk-Ulam Theorem (BUT) for topological spaces equipped with a descriptive proximity or a strong descriptive proximity.

### 2.1. Antipodal sets

In considering a string-based BUT, we consider antipodal sets instead of antipodal points. A subset of an -dimensional real linear space is a Petty antipodal set, provided, for each pair of points , there exist disjoint parallel hyperplanes such that  [29, §2]. A hyperplane is subspace of a vector space [7]. In general, a hyperplane is any codimension-1 subspace of a vector space [34].

Let be an -dimensional Euclidean space such that every member of is a string and let . are antipodal sets, provided are subsets of disjoint parallel hyperplanes. For example, strings are antipodal, provided the strings are subsets of disjoint parallel hyperplanes. In this work, we relax the Petty antipodal set parallel hyperplane requirement. That is, a pair of spatial regions (aka, strings) are antipodal, provided the regions or strings contain proper substrings of disjoint parallel hyperplanes, i.e., antipodes contain proper substrings such that .

In other words, a pair of strings are antipodal, provided there are points that belong to disjoint hyperplanes.

###### Example 1.

Let in Fig. 2 represent strings. Then

 strH,strK are antipodal, since strH∩strK=∅, strH,strE1∪strE2 are antipodal, since strH∩(strE1∪strE2)=∅, strH,strL1∪L2 are antipodal, since strH∩(strL1∪strL2)=∅, strE1,strL1 are antipodal, since strE1∩strL1=∅, strE1,strE2 are antipodal, since X∩Y≠∅, for X∈2strE1,Y∈2strE2, strL1,strL2 are antipodal, since X∩Y≠∅, for X∈2strL1,Y∈2strL2.

By definition, and are antipodal, since

 (strE1∪strE2∖strE1∩strE2) ≠∅, and (strL1∪strL2∖strL1∩strL2) ≠∅.\qquad{\color[rgb]{0,0,1}■}

In a point-free geometry [9, 10], regions (nonempty sets) replace points as the primitives. Let be a nonempty region of a space . Region is a worldsheet (denoted by ), provided every subregion of contains at least one string. In other words, a worldsheet is completely covered by strings. Let be a collection of strings. is a cover for , provided . Every member of a worldsheet is a string.

###### Example 2.

A worldsheet with finite width and finite length is represented in Fig. 3.1. This worldsheet is rolled up to form the lateral surface of a cylinder represented in Fig. 3.2, namely, with radius and height equal to the length of . We call this a worldsheet cylinder. In effect, a flattened, bounded worldsheet maps to a worldsheet cylinder.

The rolled up world sheet in Example 2 is called a worldsheet cylinder surface. Let be a pair of worldsheets. are antipodal, provided there is at least one string such that . A worldsheet cylinder maps to a worldsheet torus.

###### Example 3.

A worldsheet torus with finite radius and finite length is represented in Fig. 4. This worldsheet torus is formed by bending a worldsheet cylinder until the ends meet. In effect, a flattened, a worldsheet cylinder maps to worldsheet torus.

###### Conjecture 1.

A bounded worldsheet cylinder is homotopically equivalent to a worldsheet torus.

### 2.2. Strong Descriptive proximity

The descriptive proximity was introduced in [27]. Let and let be a feature vector for , a nonempty set of non-abstract points such as picture points. reads is descriptively near , provided for at least one pair of points, . From this, we obtain the description of a set and the descriptive intersection [17, §4.3, p. 84] of and (denoted by ) defined by

():

, set of feature vectors.

():

.

Then swapping out with in each of the Lodato axioms defines a descriptive Lodato proximity.

That is, a descriptive Lodato proximity is a relation on the family of sets , which satisfies the following axioms for all subsets of .

(dP0):

.

(dP1):

.

(dP2):

.

(dP3):

or .

(dP4):

and for each .

reads the empty set is descriptively far from . Further is descriptively separated , if

(dP5):

( and have matching descriptions).

###### Proposition 1.

[26] Let be a descriptive proximity space, . Then .

Nonempty sets in a topological space equipped with the relation , are strongly near [strongly contacted] (denoted ), provided the sets have at least one point in common. The strong contact relation was introduced in [21] and axiomatized in [23][12, §6 Appendix].

Let be a topological space, and . The relation on the family of subsets is a strong proximity, provided it satisfies the following axioms.

(snN0):

, and  .

(snN1):

.

(snN2):

implies .

(snN3):

If is an arbitrary family of subsets of and for some such that , then

(snN4):

.

When we write , we read is strongly near ( strongly contacts ). The notation reads is not strongly near ( does not strongly contact ). For each strong proximity (strong contact), we assume the following relations:

(snN5):

(snN6):

For strong proximity of the nonempty intersection of interiors, we have that or either or is equal to , provided and are not singletons; if , then , and if too is a singleton, then . It turns out that if is an open set, then each point that belongs to is strongly near . The bottom line is that strongly near sets always share points, which is another way of saying that sets with strong contact have nonempty intersection.

The descriptive strong proximity is the descriptive counterpart of . To obtain a descriptive strong Lodato proximity (denoted by dsn), we swap out in each of the descriptive Lodato axioms with the descriptive strong proximity .

Let be a topological space, and . The relation on the family of subsets is a descriptive strong Lodato proximity, provided it satisfies the following axioms.

(dsnP0):

, and  .

(dsnP1):

.

(dsnP2):

implies .

(dsnP4):

.

When we write , we read is descriptively strongly near . For each descriptive strong proximity, we assume the following relations:

(dsnP5):

.

(dsnP6):

.

###### Definition 1.

Suppose that and are topological spaces endowed with strong proximities [24]. We say that the map is strongly proximal continuous and we write s.p.c. if and only if, for ,

 A \doublewedgeδX B⇒f(A) \doublewedgeδY f(B).\qquad{\color[rgb]{0,0,0}■}
###### Theorem 2.

[24] Suppose that and are topological spaces endowed with compatible strong proximities and is s.p.c. Then is an open mapping, i.e., maps open sets in open sets.

Let be any point in that is not . The Borsuk-Ulam Theorem (BUT) has many different region-based incarnations.

###### Theorem 3.

[24] If is -continuous (s.p.c.), then for some .

###### Proof.

The mapping is s.p.c. if and only if implies . Let for some . From -Axiom (snN6), . Hence, . ∎

###### Corollary 1.

If is -continuous (s.p.c.), then for some .

Since the proof of Theorem 3 depends on the domain and range of mapping being compatible topological spaces equipped with a s.p.c. map and does not depend on the geometry of , we have

###### Theorem 4.

[24] Let and be topological spaces endowed with compatible strong proximities. If is -continuous (s.p.c.), then for some .

###### Corollary 2.

Let be a nonempty set of strings. If is -continuous (s.p.c.), then for some .

## 3. String-Based Borsuk Ulam Theorem (strBUT)

This section considers string-based forms of the Borsuk Ulam Theorem (strBUT). Recall that a region is limited to a set in a metric topological space. By definition, a string on the surface of an -sphere is a line that represents the path traced by moving particle along the surface of the . Disjoint strings on the surface of that are antipodal and with matching description are descriptively near antipodal strings. A pair of strings are antipodal, provided, for some , there exist disjoint parallel hyperplanes such that and . Such strings can be spatially far apart and also descriptively near.

###### Example 4.

A pair of antipodal strings are represented by in Fig. 5. Strings are antipodal, since they have no points in common. Let bounded shape be a feature of a string. A shape is bounded, provided the shape is surrounded (contained in) another shape. Then are descriptively near, since they are both bounded shapes.

We are interested in the case where strongly near strings are mapped to strongly near strings in a feature space. To arrive at a string-based form of Theorem 4, we introduce region-based strong proximal continuity. Let be a nonempty set and let denote the family of all subsets in . For example, is the family of all subsets on the surface of an -sphere.

###### Definition 2.

[22, §5.7].
Let be nonempty sets. Suppose that and are topological spaces endowed with strong proximities. We say that is region strongly proximal continuous and we write Re.d.p.c. if and only if, for ,

 A δΦ B⇒f(A) δΦ f(B).\qquad{\color% [rgb]{0,0,1}■}
###### Lemma 1.

[24] Suppose that and are topological spaces endowed with compatible descriptive proximities and is a Re.d.p.c. continuous mapping on the family of regions into . If is a description common to antipodal regions , then for some .

Lemma 1 is restricted to regions in described by feature vectors in an -dimensional Euclidean space . Next, consider regions on the surface of an -sphere . Each feature vector in describes a region . Then we obtain the following result.

###### Theorem 5.

[24] Suppose that and are topological spaces endowed with compatible strong proximities. Let , a region in the family of regions in . If is Re.d.p.c. continuous, then for antipodal region .

###### Theorem 6.

Suppose that and are topological spaces endowed with compatible strong proximities. Let , a string in the family of strings in . If is Re.d.p.c. continuous, then for antipodal string .

###### Proof.

Let each string be a spatial subregion of . Let . Swap out with in Theorem 5 and the result follows. ∎

###### Theorem 7.

Suppose that and are topological spaces endowed with compatible strong proximities. Let be a worldsheet in the family of worldsheets in . If is Re.d.p.c. continuous, then for antipodal .

###### Proof.

The proof is symmetric with the proof of Theorem 6. ∎

###### Remark 1.

Theorem 6 and Theorem 7 are the simplest forms of string-based Borsuk-Ulam Theorem (strBUT). In this section, we also consider other forms of strBUT that arise naturally from strong forms of descriptive proximity and which have proved to be useful in a number of applications.

###### Definition 3.

Region-Based -Continuous Mapping [22, §5.7].
Let be nonempty sets. Suppose that and are topological spaces endowed with strong proximities. We say that is region strongly proximal continuous and we write Re.s.p.c. if and only if, for ,

 A \doublewedgeδX B⇒f(A) \doublewedgeδY f(B).\qquad{\color[rgb]{0,0,1}■}

For an introduction to s.p.c. mappings, see [25]. Let . For a Re.s.p.c. mapping on the collection of subsets into , the assumption is that is a region-based object space (each object is represented by a nonempty region) and is a feature space (each region in maps to a feature vector in an -dimensional Euclidean space such that is a description of region that matches the description of ).

###### Definition 4.

Region-Based -Continuous Mapping.
The mapping is region -continuous and we write Re.d.s.p.c. if and only if, for ,

 A \doublewedgeδΦ B⇒f(A) \doublewedgeδΦ f(B),

where is a feature vector that describes region .

###### Lemma 2.

[28].
Suppose that and are topological spaces endowed with compatible strong descriptive proximities and is a continuous mapping on the family of regions into . If is a description common to antipodal regions , then for some .

Lemma 2 is restricted to regions in described by feature vectors in an -dimensional Euclidean space . Next, consider regions on the surface of an -sphere . Each feature vector in describes a region . Then we obtain the following result.

###### Theorem 8.

[28].
Suppose that and are topological spaces endowed with compatible strong proximities. Let , a region in the family of regions in . If is -Re.s.p continuous, then for region .

###### Theorem 9.

Suppose that and are topological spaces endowed with compatible strong proximities. Let be a worldsheet in the family of regions in . If is -Re.s.p continuous, then for region .

###### Proof.

Since a worldsheet is a region in , the result follows from Theorem 8. ∎

###### Example 5.

The -Re.s.p continuous mapping is represented in Fig. 6. In this example, it is assumed that the worldsheets have matching descriptions one feature, e.g., area. In that case, .

In the proof of Theorem 8 and Theorem 9, we do not depend on the fact that each region is on the surface of a hypersphere . For this reason, we are at liberty to introduce a more general region-based Borsuk-Ulam Theorem (denoted by reBUT), applicable to strings and worldsheets.

Let . For a Re.s.p.c. mapping on the collection of subsets to , the assumption is that is a region-based object space (each object is represented by a nonempty region) and is a feature space (each region in maps to a feature vector in such that is a description of region ). Then we obtain the following result

###### Theorem 10.

[28].
Suppose that and are topological spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some .

###### Theorem 11.

strBUT for Worldsheets.
Suppose that and are topological spaces endowed with compatible strong proximities, . If is -Re.s.p continuous, then for some .

###### Proof.

Let each worldsheet be a spatial subregion of . Let . Swap out with in Theorem 10 and the result follows. ∎

###### Example 6.

Let in Fig. 6 represent a pair of antipodal worldsheets. For simplicity, we consider only the feature worldsheet area. Let map a worldsheet to a real number that is the area of , i.e., equals the area of . It is clear that more than one antipodal worldsheet will have the same area. Then, from Theorem 11, for some .

###### Lemma 3.

Region Descriptions in a -Dimensional Space.
Suppose that and are topological spaces endowed with compatible strong proximities and on the family of regions maps into . If is a description common to antipodal regions , then in for some in for .

###### Proof.

Let . The assumption is that is a feature vector in a -dimensional feature space that describes as well as at least one other region in . We consider only the case for for shape-connected regions that are disks in a finite, bounded, rectangular shaped space in the Euclidean plane, where every region has 4,3 or 2 adjacent disks, e.g., each corner region has at most 2 adjacent disks. Let

 f(A)={2,if A is a corner region with 2 adjacent polygons,level>2,otherwise.

Let be antipodal corner regions. Then in . ∎

Lemma 3 leads to a version of reBUT for region descriptions in a -dimensional feature space.

###### Theorem 12.

Proximal Region-Based Borsuk-Ulam Theorem (rexBUT).
Suppose that and are topological spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some .

###### Proof.

Swap out with in the proof of Theorem 10 and the result follows. ∎

A string space is a nonempty set of strings. A worldsheet space is a nonempty set of worldsheets.

###### Corollary 3.

Let be a string space. Assume , , are topological string spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some string .

###### Corollary 4.

Let be a worldsheet space. Assume , , are topological worldsheet spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some worldsheet .

### 3.1. String Space and Worldsheet Space

A string space is a nonempty set of strings. A worldsheet space is a nonempty set of worldsheets.

###### Corollary 5.

Let be a string space. Assume , , are topological string spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some string .

###### Corollary 6.

Let be a worldsheet space. Assume , , are topological worldsheet spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some worldsheet .

Let be a region in the family of sets , a feature vector with components that describes region . A straightforward extension of Theorem 12 leads to a continuous mapping of antipodal regions in an -dimensional space in to regions in a -dimensional feature space .

###### Theorem 13.

Region-2-Region Based Borsuk-Ulam Theorem (re2reBUT).
Suppose that , where space is -dimensional and are topological spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some .

###### Proof.

From Lemma 3 and swapping out with in the proof of Theorem 12, the result follows. ∎

From Theorem 13, we obtain the following results relative to strings and worldsheets.

###### Corollary 7.

Let be a string space. Assume , , are topological string spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some set of strings .

###### Corollary 8.

Let be a worldsheet space. Assume , , are topological worldsheet spaces endowed with compatible strong proximities. If is -Re.s.p continuous, then for some set of worldsheets .

###### Example 7.

Let the Euclidean spaces and be endowed with the strong proximity and let be antipodal strings in . Further, let be a proximally continuous mapping on into defined by

 strA ∈2S2, Φ :2S2⟶R3, defined by Φ(strA) =(bounded,finite,length)∈R3, and f:2S2 ⟶2R4, defined by f(2S2) ={Φ(strA)∈R3:strA∈2S2}∈2R4.\qquad{\color[rgb]{0,0,1}■}

### 3.2. Brouwer’s Fixed Point Theorem and Wired Friend Theorem

Next, consider Brouwer’s fixed point theorem [2, 3, 4, 5]. Let denote the unit -ball in , which is the interior of a sphere . Let be a point in and let . Then a closed unit ball [16, §24] is defined by

 Bn={x:∥x∥≤1}.

A fixed point for a map is a point so that .

###### Theorem 14.

Brouwer’s Fixed Point Theorem for [2] Let and let be a closed unit ball in . Then every continuous map has a fixed point.

###### Proof.

The proof for and closed unit ball in is given in [30, §5, Theorem 5.1.22]. J. Hadamard gave a proof of this theorem for all dimensions [13]. ∎

A complete study of Brouwer’s Fixed Point Theorem is given by T. Stuckless [32].

###### Example 8.

Coffee Cup Illustration of Brouwer’s Fixed Point Theorem for Dimension 3.
F.E. Su [33] gives a coffee cup illustration of the Brouwer Fixed Point Theorem (our Theorem 14) for all dimensions. No matter how you continuously slosh the coffee around in a coffee cup, some point is always in the same position that it was before the sloshing began. And if you move this point out of its original position, you will eventually move some other point in the sloshing coffee back into into its original position.

###### Example 9.

Proof of Brouwer’s Fixed Point Theorem for Dimension .
F.E. Su [33] gives a constructive proof of the Brouwer Fixed Point Theorem (our Theorem 14) for all dimensions. V. Runde also gives a beautiful proof of Brouwer’s Theorem for dimensions.

We can always find a ball containing , which is a description of a string with features on a hypersphere . Also, observe that each has a particular shape (denoted by ).

###### Remark 2.

String Theory and Wired Friends.
The shape of a string is the silhouette of a string . A is called a wired friend of an object . Every wired friend is known by its shape.

These observations lead to a wired friend theorem. The proof of Theorem 15 uses a projection mapping. Let be an object space, a set of shapes in an -dimensional space derived from and a -dimensional feature space with . Each component of a feature vector in is a feature value of a shape in . Let read maps to. Then a projection mapping is defined by

 O ⟼ X ⟼ Y.
###### Example 10.

Sample Projection Mapping a Set of 2-dimensional (flat) Worldsheets.
.
Let be a flat 2D worldsheet is a set of world sheets . Let be a cylinder (rolled up worldsheet). And let be a set of -dimensional worldsheet cylinders (flat worldsheets rolled up into a cylinder). And let be a -dimensional feature space . Each component of a feature vector in is a feature value of a cylinder . Then a projection mapping is defined by

 {wshA} f⟼ {cylA} g⟼ Rk

Define . Select () features of a cylinder . Then define

 g(f(wshA))=g(cylA)=k−dimensional feature vector in Rk.

###### Theorem 15.

Wired Friend Theorem
Every occurrence of a wired friend with a particular strShape with features on maps to a fixed description that belongs to a ball in .

###### Proof.

From Theorem 6, we know in for every on or, for that matter (from Theorem 13), in any space containing . Every friend can be represented by a set of strings. Observe that is a fixed description of a of a wired friend . In other words, we have a projection mapping:

 strA f⟼ strShapeA g⟼ description g(strShapeA),

i.e., a wired friend maps to string shape such that . And maps to a description of the string shape, namely . Since every belongs to a -dimensional ball