Hausdorff Spaces in Bermejo Algebras: The Birth of Treonic Manifold Construction

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Hausdorff Spaces in Bermejo Algebras: The Birth of Treonic Manifold Construction

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Alejandro Jesús Bermejo Valdés^*

This version is not peer-reviewed

Submitted:

10 July 2024

Posted:

11 July 2024

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Abstract

We conducted a topological analysis of the novel, non-associative, and unital algebraic structure known as Bermejo Algebras, developed by Alejandro Bermejo, which includes the Algebra B and Treon Algebra. This algebras can define Lie and Malcev algebras when their product operations are derived from the Bermejo Algebras product. Central to this study are treons, complex entities that arise as isomorphisms of Algebra B when the field is real. We define the vector spaces associated with Bermejo Algebras to establish equivalence classes and a quotient topology, resulting in Hausdorff spaces that do not depend on traditional norms, inner products, or metrics. Our findings lay the groundwork for constructing new types of differential manifolds, opening new avenues for advanced mathematical research.

Keywords:

Subject: Computer Science and Mathematics - Geometry and Topology

1. Introduction

Bermejo Algebras are a novel non-associative and unital algebraic structure, founded and developed by Alejandro Bermejo [1,2]. This algebraic construction can define Lie and Malcev algebras when their product operations are derived from the product operation of Bermejo Algebras. These algebras yield complex entities, such as the algebra

C^{2}

of vectors

(a_{1}, a_{2})

, which can be considered a trivial case of Bermejo Algebra:

(a_{1}, a_{2}, 0)

. These entities are distinct from quaternions and other higher hypercomplex structures [2]. Bermejo termed these complex entities "treons", which manifest as isomorphisms of his algebra when the field is real (

R

) [2].

We undertake a topological analysis of Bermejo Algebras, redefining the fundamental space of these algebras to develop a structure based solely on the products of Bermejo Algebras. By avoiding the need to define norms, inner products, or metrics, we successfully establish equivalence classes and a quotient topology that forms a Hausdorff space. Given that Hausdorff spaces are crucial in the definition of manifolds [3,4], our work lays the essential groundwork for constructing treonic differential manifolds.

This work marks the beginning of constructing a framework to support any complex analysis of the spaces discussed here. For instance, the Cauchy-Riemann equations for treons [5] will be better defined with the vector bases established here.

2. Bermejo Algebras

Bermejo algebras (Algebra B [1] and Treon Algebra [2]) can be expressed both as points in

R^{3}

, for the case

(a_{1}, a_{2}, a_{3})

, or as points in a 3-dimensional "complex space

C^{3}

", for the case

a_{1} + a_{2} i + a_{3} j

The algebra B isomorphic to the treon algebra necessarily implies the definition of algebra B over the real field

R

; therefore, the algebra B isomorphic to the treonic algebra uses

R^{3}

as a vector space and associates it with a product. Consequently, algebra B isomorphic to treon algebra is an algebra over

R^{3}

and, logically, treon algebra is an algebra over

R^{3}

[2].

Now, if we assume that in Bermejo algebras there are complex quantities [1,2,5], we can define a vector space with a subtly different base structure, i.e., the canonical basis of

R^{3}

can be defined from bases in a space we will call B-Spaces:

Let

{\hat{i}, \hat{j}, \hat{k}}

be the canonical base in

R^{3}

, we define:

\hat{i} \equiv (1, 0, 0), \hat{j} \equiv (0, 1, 0), \hat{k} \equiv (0, 0, 1)

where

{(1, 0, 0), (0, 1, 0), (0, 0, 1)}

is the basis for the space associated with algebra B [1,2]: Space

B_{1}

. And:

(1, 0, 0) \equiv id, (0, 1, 0) \equiv i, (0, 0, 1) \equiv j

where

{id, i, j}

is the basis in the space associated with treon algebra [2]: Space

B_{2}

. And, by transitivity of the equivalence ≡:

\hat{i} \equiv id, \hat{j} \equiv i, \hat{k} \equiv j .

In this way, space

B_{1}

will be generated by the span

((1, 0, 0), (0, 1, 0), (0, 0, 1))

and space

B_{2}

will be generated by the span

{id, i, j}

, such that the linear independence of both bases is inherited from the linear independence of the canonical base in

R^{3}

Since

R^{3}

is a differentiable manifold, B-spaces are a differentiable manifold. However, these spaces are subject to and limited by the topology induced by the Euclidean metric. In this work, we carry out a step-by-step analysis of the properties of B-spaces to define a differentiable manifold not limited by the Euclidean metric, but at the same time influenced and defined by a particular property of Bermejo Algebras, the appearance of a "norm" in the real component of the double conjugated square of an element. We cannot define the squared norm of a treon in the conventional sense without introducing an inner product or a metric in the vector space. Here we present a pioneering analysis that allows working with norms, metrics, and inner products in a non-normed, non-metric space without inner products; at least not from the conventional viewpoint. This is because the products of Bermejo Algebras naturally produce these properties.

3. Construction of a Topology on Treons

3.1. Definitions for Treonic Topology

To construct a topology T on the space of treons,

B_{2}

, which we denote X for simplicity, we define:

Let X be the set of treonic elements

p \equiv p_{1} id + p_{2} i + p_{3} j

, with

i^{2} = j^{2} = - id

, where id is the identity element of the algebra. To simplify, we will assume from here that

id = 1

. Let

P (X)

be the power set of X, and let a topology

T \subseteq P (X)

. By definition, T must satisfy [6,7]:

$\emptyset, X \in T$ , where ∅ represents the empty set.
If $U_{1}, U_{2} \in T$ , then $U_{1} \cap U_{2} \in T$ (finite intersection), where $U \in T$ are the open sets of topology T.
If ${(U_{i})}_{i \in I} \in T$ , then $⋃_{i \in I} U_{i} \in T$ . The index I is an arbitrary index, finite or not.

By definition, we have two trivial topologies: the indiscrete treonic topology,

T = {\emptyset, X}

, and the discrete treonic topology,

T = P (X)

Let the pair

(X, T)

be our treonic topological space, we say that V is an open neighborhood of a treon p if:

V \in T \land p \in V

3.2. Convergence of Treons and Definition of Hausdorff Space

We define a sequence of treons as a mapping

φ

such that:

φ : N \to X,

n \mapsto φ (n) \equiv p_{{n}},

where

{n}

is a notation to represent the successive elements:

p_{{n}} = p_{1 {1}} + p_{2 {1}} i + p_{3 {1}} j, p_{1 {2}} + p_{2 {2}} i + p_{3 {2}} j, p_{1 {3}} + p_{2 {3}} i + p_{3 {3}} j, \dots

We say that a sequence of treons

p_{{n}}

converges to an element p, and we denote

p_{{n}} \to p

, if and only if:

\forall V, p \in V, \exists N \in N : \forall n \geq N \Rightarrow p_{{n}} \in V .

The topological space

(X, T)

is called a Hausdorff space if and only if [3,4]:

\forall p_{a}, p_{b} \in X, p_{a} \neq p_{b}, \exists V_{p_{a}} \in T \land V_{p_{b}} \in T : V_{p_{a}} \cap V_{p_{b}} = \emptyset,

where

V_{p_{a}}

is the open neighborhood of

p_{a}

, and

V_{p_{b}}

is an open neighborhood of

p_{b}

Since

R^{3}

is a Hausdorff space with the topology induced by the Euclidean metric [3,4], both space

B_{1}

and space

B_{2}

are Hausdorff spaces.

A Hausdorff space guarantees that the limits of sequences of points are unique [3,4]. If a sequence

p_{{n}}

of treons converges to both

p_{a}

and

p_{b}

, necessarily

p_{a} = p_{b}

. In a Hausdorff space, a sequence that converges to two different points generates a contradiction.

If we have a Hausdorff space and if

p_{{n}}

converges to both

p_{a}

and

p_{b}

we have:

p_{{n}} \to p_{a} \Leftrightarrow \forall V_{a}, p_{a} \in V_{a}, \exists N_{a} \in N : \forall n \geq N_{a} \Rightarrow p_{{n}} \in V_{a},

p_{{n}} \to p_{b} \Leftrightarrow \forall V_{b}, p_{b} \in V_{b}, \exists N_{b} \in N : \forall n \geq N_{b} \Rightarrow p_{{n}} \in V_{b} .

Therefore, taking simultaneously the maximum

(N_{a}, N_{b})

, we will have that

p_{{n}} \in V_{a}

and

p_{{n}} \in V_{b}

. This contradicts the definition of a Hausdorff space, where there is no point

p_{{n}}

in the intersection

V_{a} \cap V_{b}

Hausdorff spaces are a fundamental pillar in the definition of manifolds [3,4,8,9]; therefore, in this work, we develop the elementary mathematical tools for defining Hausdorff spaces under the analysis of Bermejo Algebras. This is the first step in generating treonic manifolds.

3.3. Definition of Treonic Quotient Topology

Let

(X, T)

be our treonic topological space, and let ∼ be an arbitrary equivalence relation on X; let

{[p]}_{\sim}

be an equivalence class for

p_{i} \in X

under the relation ∼, and let

X / \sim

be the quotient set. We define the mapping (surjective) canonical projection on the set of treons X as:

q : X \to X / \sim,

p \mapsto {[p]}_{\sim} .

We define a preimage of the mapping q, which we denote

{Preim}_{q} \subseteq X

Let an arbitrary

{Preim}_{q} (U) \subseteq X

, such that

{Preim}_{q} (U) \in T

, the canonical projection mapping q implies, by definition, the obligatory existence of an open set

U \subseteq X / \sim

, such that

U \in \tilde{T}

, where

\tilde{T}

is a new topology defined on

X / \sim

, which is simply a quotient topology [10] for the case of treons. We call the pair

(X / \sim, \tilde{T})

the treons quotient space.

3.4. Analysis of the Real Component of $p ⊙ p^{((*_{i, j}))}$

In algebra B, we define

〈 p^{2} 〉 \equiv p ⊙ p^{((*_{i, j}))}

, where ⊙ is the product of algebra B, and

(*_{i, j})

is the double complex conjugation of a treon, i.e., a conjugation in i and a conjugation in j [1,2]. According to this,

〈 p^{2} 〉 = {(∥ p ∥}^{2}, 2 p_{1} p_{2} + p_{2} p_{3}, 2 p_{1} p_{3} + p_{3} p_{2})

, where:

{∥ p ∥}^{2} \equiv p_{1}^{2} + p_{2}^{2} + p_{3}^{2} .

The operation

〈 p^{2} 〉 \equiv p ⊙ p^{((*_{i, j}))}

we name it "double conjugated square". Bermejo called

{∥ p ∥}^{2}

the "norm squared", but it should not be interpreted as the norm squared per se of p, since only the real part of

〈 p^{2} 〉 \in X

is being considered. Note that this "norm squared" is the real part of the double conjugated square of a treon, i.e.,

{∥ p ∥}^{2} \equiv Re (〈 p^{2} 〉) .

We define the subset

N \subset X

, such that:

N \equiv {p \in X : p = 〈 p^{2} 〉},

and the mapping

〈 \cdot 〉

〈 \cdot^{2} 〉 : X \to N \subset X,

p \mapsto 〈 p^{2} 〉 .

{∥ p ∥}^{2} \equiv Re (〈 p^{2} 〉)

implies that the real part of a treon

〈 p^{2} 〉 \in N

is a place where Euclidean norms of three components of a real vector

\vec{p} \equiv p_{1} \hat{i} + p_{2} \hat{j} + p_{3} \hat{k}

arise, whose components coincide with the components of the treon

p = p_{1} + p_{2} i + p_{3} j

, an element of the domain of

〈 \cdot^{2} 〉

. We understand that the norm of a real vector arises in the real component of a treon when performing the double conjugated square.

Note that

{∥ p ∥}^{2} = 0 \Rightarrow p = 0 \land 〈 p^{2} 〉 = 0

. Therefore, within the set where norms are defined (the set N), any treon

〈 p^{2} 〉

with a zero real part will be a zero treon, and its preimage under

〈 \cdot^{2} 〉

will be the zero treon in X. Therefore, in N, any pure imaginary treon

p_{2} i + p_{3} j

is zero and derives from a zero treon; this means that there are no pure imaginary treons in N. This does not occur in

X ∖ N

, where a pure imaginary treon produces an element

〈 p^{2} 〉 = (p_{2}^{2} + p_{3}^{2}, p_{2} p_{3}, p_{3} p_{2}) \in N

Theorem: In N, the norm of a treon is zero if and only if the treon is zero:

\forall p \in N \subset X : Re (p) = 0 \Leftrightarrow p = 0 + 0 i + 0 j .

Proof ⇒:

We have that

〈 p^{2} 〉 = {(∥ p ∥}^{2}, 2 p_{1} p_{2} + p_{2} p_{3}, 2 p_{1} p_{3} + p_{3} p_{2})

, such that

{∥ p ∥}^{2} \equiv p_{1}^{2} + p_{2}^{2} + p_{3}^{2}

. If

{∥ p ∥}^{2} = 0 \Leftrightarrow p_{1} = 0 \land p_{2} = 0 \land p_{3} = 0

, then

p = p_{1} + p_{2} i + p_{3} j = 0 + 0 i + 0 j

Proof ⇐:

p = 0 + 0 i + 0 j

, then

p_{1} = 0

. Therefore,

Re (p) = 0

□

Taking into account the product ⊙ of algebra B [1], defined as:

p_{A} ⊙ p_{B} \equiv (p_{A 1} p_{B 1} - p_{A 2} p_{B 2} - p_{A 3} p_{B 3}, p_{A 1} p_{B 2} + p_{A 2} p_{B 1} + p_{A 3} p_{B 2}, p_{A 1} p_{B 3} + p_{A 3} p_{B 1} + p_{A 3} p_{B 2}),

we have:

p_{A} ⊙ p_{B}^{((*_{i, j}))} = (p_{A 1} p_{B 1} + p_{A 2} p_{B 2} + p_{A 3} p_{B 3}, - p_{A 1} p_{B 2} + p_{A 2} p_{B 1} - p_{A 3} p_{B 2}, - p_{A 1} p_{B 3} + p_{A 3} p_{B 1} - p_{A 3} p_{B 2}),

where we will define

Re (p_{A} ⊙ p_{B}^{((*_{i, j}))}) \equiv Re (〈 p_{A}, p_{B} 〉) \equiv p_{A} ⋄ p_{B} \in R

, which, while it has the structure of an inner product, we do not define it as such. The diamond operation ⋄ naturally arises from the definition of the product of algebra B. Therefore, we have:

〈 p_{A}, p_{B} 〉 = (p_{A} ⋄ p_{B}, p_{A 2} p_{B 1} - p_{A 1} p_{B 2} - p_{A 3} p_{B 2}, p_{A 3} p_{B 1} - p_{A 1} p_{B 3} - p_{A 3} p_{B 2}) .

Then, with this, we have an operation involving the product of a treon with another doubly conjugated treon, resulting in treons with a real component exhibiting the structure of an "inner product". We refer to the operation

〈 p_{A}, p_{B} 〉

as the Bermejian inner product. In this context, the case of the doubly conjugated square is considered a particular instance of the Bermejian inner product, specifically for the product of identical treons.

On the other hand,

{∥ p ∥}^{2} = 0 \Rightarrow \vec{p} = \vec{0}

. Therefore, within N, any zero treon maps to the zero vector

\vec{p} = \vec{0}

R^{3}

through a mapping h:

h : N \to R^{3},

〈 p^{2} 〉 \mapsto \vec{p} .

Taking the set of all treons

p \in X

that have an image in

N \subset X

under

〈 \cdot^{2} 〉

{Preim}_{〈 \cdot^{2} 〉}

, we can define a composition mapping

H = h \circ 〈 \cdot^{2} 〉

, such that:

H : {Preim}_{〈 \cdot^{2} 〉} \to R^{3},

p \mapsto \vec{p} .

This mapping ensures that all elements have a defined norm, such that the only possibility for an element to have a zero norm is for the element to be zero

(0, 0, 0)

Note that the components of

\vec{p}

in the canonical basis, as defined, can be made to coincide with the components of a treon

p \in {Preim}_{〈 \cdot^{2} 〉}

in its corresponding base

{id, i, j}

. In this way:

\forall \vec{p} \in R^{3}, \forall p \in {Preim}_{〈 \cdot^{2} 〉}, \exists ∥ \vec{p} ∥ \in R : ∥ \vec{p} ∥ = \sqrt{Re (〈 p^{2} 〉)} .

The vectors

\vec{p} \in R^{3}

that are a representation of the treons

p \in {Preim}_{〈 \cdot^{2} 〉}

under the composition mapping H, we denote as

ρ

3.5. $S_{r}^{2}$ -Spheres and Treonic Equivalence Classes

We define an

S_{r}^{2}

-sphere as the set of vectors

ρ

, such that

Re ({〈 p^{2} 〉}_{i}) = r^{2} \in R

S_{i}^{2} \equiv {ρ \in R^{3} : \sqrt{Re ({〈 p^{2} 〉}_{i})} = r_{i} \land r_{i} > 0},

where

S_{i}^{2}

are the different spheres of radius

r_{i}

greater than zero. This notation, in the traditional sense, refers to 2-spheres of radius r. For the purposes of our algebra, we have renamed this as

S^{2} r

-spheres.

With this, we construct an equivalence relation for the elements

p \in {Preim}_{〈 \cdot^{2} 〉}

. Let the following equivalence relation ∼:

\forall p_{i}, p_{j} \in {Preim}_{〈 \cdot^{2} 〉} \subset X, \forall ρ_{i}, ρ_{j} \in R^{3} : p_{i} \sim p_{j} \Leftrightarrow ρ_{i} = ρ_{j} ⊻ ρ_{i} = - ρ_{j},

where ⊻ denotes the binary logical operator "exclusive or", which excludes the truth value "True" when both propositions are True.

Proof that $p_{i} \sim p_{j}$ is an equivalence relation:

1. Reflexivity: We have

p_{i} \sim p_{i} \Leftrightarrow ρ_{i} = ρ_{i} ⊻ ρ_{i} = - ρ_{i}

. Clearly, this proposition is true because

ρ_{i} = 0

imply null norms and, therefore, null vectors and treons that do not conform to the sphere by definition.

2. Symmetry:

p_{i} \sim p_{j} = p_{j} \sim p_{i}

. Clearly, this holds since

ρ_{i} = ρ_{j}

is the same as

ρ_{j} = ρ_{i}

, and since

ρ_{i} = - ρ_{j}

is the same as

ρ_{j} = - ρ_{i}

3. Transitivity: For

p_{i} \sim p_{k}

and

p_{k} \sim p_{j}

we have:

p_{i} \sim p_{k} \Leftrightarrow ρ_{i} = ρ_{k} ⊻ ρ_{i} = - ρ_{k},

p_{k} \sim p_{j} \Leftrightarrow ρ_{k} = ρ_{j} ⊻ ρ_{k} = - ρ_{j},

ρ_{i} = ρ_{k} = ρ_{j}

, and as

ρ_{i} = - ρ_{k} = - (- ρ_{j}) = ρ_{j}

, we have that the proposition

p_{i} \sim p_{j}

is true as it verifies the unique possibility of transitivity that

ρ_{i} = ρ_{j}

. Thus, transitivity verifies

p_{i} \sim p_{j} \Leftrightarrow ρ_{i} = ρ_{j}

only, being false

ρ_{i} = - ρ_{j}

□

We extended the definition of

S_{r}^{2}

-spheres to the treonic set X. We define:

Λ \equiv {p \in {Preim}_{〈 \cdot^{2} 〉} \subset X : \sqrt{Re ({〈 p^{2} 〉}_{i})} = r_{i} \land r_{i} > 0} .

This is simply a change in the way of representing:

ρ

is a vector on the canonical base while p is a treon on the treonic base

{id, i, j}

We call the set

Λ

the "r-treosphere," again, a change in the way of thinking about the spaces where they are defined. For

r = 1

we have a 1-treosphere such that there is an equivalence class

{[p]}_{\sim_{(r = 1)}}

, for

r = 2

we have

{[p]}_{\sim_{(r = 2)}}

, and for any

r = r_{0} \in R

, we have

{[p]}_{\sim_{R}}

. The set

{[p]}_{\sim_{R}}

for any

r_{0} \in R

allows grouping the treons according to these equivalence classes.

We define

Λ / \sim

as the set of all

{[p]}_{\sim_{R}}

in X. And we construct the treons quotient space

(Λ / \sim, \tilde{T})

. Note that

{[p]}_{\sim_{R}}

involves all the points in the volume of a sphere.

We denoted the topology T of

(X, T)

that is in

Λ

T_{Λ}

, and the corresponding topological subspace as

(Λ, T_{Λ})

. Thus,

(Λ, T_{Λ})

is a topological subspace of

(X, T)

The Treons Quotient Space is a Hausdorff Space

Let two treons

p_{i}

and

p_{j} \in Λ

, such that

ρ_{i} \neq ρ_{j} \land ρ_{i} \neq - ρ_{j}

, then

{[p_{i}]}_{\sim} \neq {[p_{j}]}_{\sim}

. Two treons

p_{i}

and

p_{j}

are equal if their corresponding components are equal; therefore, if

ρ_{i} \neq ρ_{j}

and

ρ_{i} \neq - ρ_{j}

, then

p_{i} \neq p_{j}

. On the other hand, the canonical projection mapping

q \circ h \circ 〈 \cdot^{2} 〉 : Λ \to Λ / \sim

is surjective, not injective, and we can have two treons

p_{i}

and

p_{j}

mapped to an equivalence class

{[p]}_{\sim}

, i.e., we can have two treons

p_{i}

and

p_{j}

such that

ρ_{i} = ρ_{j}

⊻

ρ_{i} = - ρ_{j}

. This implies that the construction of a neighborhood in

Λ / \sim

implies the definition of two neighborhoods in

S_{i}^{2}

and, by extension, in

Λ

We define the following treons:

p_{(ζ, r_{0})} \equiv {p_{ζ} \in Λ : ρ_{ζ} \Rightarrow \sqrt{Re (〈 p_{ζ}^{2} 〉)} = r_{0} > 0},

p_{(- ζ, r_{0})} \equiv {p_{(- ζ)} \in Λ : ρ_{(- ζ)} = - ρ_{ζ}},

where

p_{(ζ, r_{0})}

and

p_{(- ζ, r_{0})}

are the two elements of an equivalence class

{[p_{ζ}]}_{\sim_{(r = r_{0})}} \in Λ / \sim

Thus, we have:

H : Λ \to R^{3},

p_{(ζ, r_{0})} \mapsto ρ_{(ζ, r_{0})},

p_{(- ζ, r_{0})} \mapsto ρ_{(- ζ, r_{0})} .

Let the topological space

(Λ, T_{Λ})

, we can take a

p_{(ζ, r_{0})} \in V_{(ζ, r_{0})}

such that

V_{(ζ, r_{0})} \in T_{Λ}

; therefore,

V_{(ζ, r_{0})}

is an open neighborhood in

(Λ, T_{Λ})

. This can be done for any r and for any pair

p_{ζ}, p_{(- ζ)} \in Λ

. By definition of topology,

V_{(ζ, r)} \cup V_{(- ζ, r)} \in T_{Λ}

Additionally, let the topological spaces

(Λ, T_{Λ})

and

(Λ / \sim, {\tilde{T}}_{Λ})

, a set

V \subset Λ / \sim

is said to be open in

{\tilde{T}}_{Λ}

if and only if

{Preim}_{q} (V) \in T_{Λ}

, which is precisely the definition of continuity between topological spaces. Therefore,

{Preim}_{q} (V_{ζ}) = V_{(ζ, r)} \cup V_{(- ζ, r)}

implies that the canonical projection q is continuous, i.e., it implies the existence of an open

V_{ζ} \in {\tilde{T}}_{Λ}

{\tilde{T}}_{Λ}

is the quotient topology of the quotient space

(Λ / \sim, {\tilde{T}}_{Λ})

We can choose neighborhoods

V_{(μ, r_{0})} \in T_{Λ}, V_{(- μ, r_{0})} \in T_{Λ}, V_{(η, r_{0})} \in T_{Λ},

and

V_{(- η, r_{0})} \in T_{Λ}

, as small as we want; therefore, we can assume the following intersections:

\begin{matrix} V_{(μ, r_{0})} \cap V_{(η, r_{0})} & = \emptyset, \\ V_{(- μ, r_{0})} \cap V_{(- η, r_{0})} & = \emptyset, \\ V_{(- μ, r_{0})} \cap V_{(η, r_{0})} & = \emptyset, \\ V_{(μ, r_{0})} \cap V_{(- η, r_{0})} & = \emptyset, \\ V_{(η, r_{0})} \cap V_{(- η, r_{0})} & = \emptyset, \\ V_{(- μ, r_{0})} \cap V_{(μ, r_{0})} & = \emptyset . \end{matrix}

Using

{Preim}_{q} (V_{ζ}) = V_{(ζ, r)} \cup V_{(- ζ, r)}

, we have:

{Preim}_{q} (V_{η}) = V_{(η, r_{0})} \cup V_{(- η, r_{0})},

{Preim}_{q} (V_{μ}) = V_{(μ, r_{0})} \cup V_{(- μ, r_{0})},

such that

{Preim}_{q} (V_{η}) \in T_{Λ}, {Preim}_{q} (V_{μ}) \in T_{Λ}, V_{η} \in {\tilde{T}}_{Λ},

and

V_{η} \in {\tilde{T}}_{Λ}

Evaluating

{Preim}_{q} (V_{η} \cap V_{μ})

, we have:

\begin{matrix} {Preim}_{q} (V_{η} \cap V_{μ}) & = {Preim}_{q} (V_{η}) \cap {Preim}_{q} (V_{μ}) \\ = (V_{(η, r_{0})} \cup V_{(- η, r_{0})}) \cap (V_{(μ, r_{0})} \cup V_{(- μ, r_{0})}) \\ = ((V_{(η, r_{0})} \cup V_{(- η, r_{0})}) \cap V_{(μ, r_{0})}) \cup ((V_{(η, r_{0})} \cup V_{(- η, r_{0})}) \cap V_{(- μ, r_{0})}) \\ = (V_{(η, r_{0})} \cap V_{(μ, r_{0})}) \cup (V_{(- η, r_{0})} \cap V_{(μ, r_{0})}) \cup (V_{(η, r_{0})} \cap V_{(- μ, r_{0})}) \cup (V_{(- η, r_{0})} \cap V_{(- μ, r_{0})}) \\ = \emptyset . \end{matrix}

Therefore,

{Preim}_{q} (V_{η} \cap V_{μ}) = \emptyset

Since the mapping q is surjective, each element of the codomain has at least one preimage in the domain. If we start from the premise

{Preim}_{q} (V_{η} \cap V_{μ}) = \emptyset

, this implies that there is no element in

Λ

such that

q (p \in Λ) \in V_{η} \cap V_{μ}

, i.e., there is no element in the domain of q that is mapped to the intersection

V_{η} \cap V_{μ}

. Consequently, due to surjectivity, for there to be no

p \in Λ

such that

q (p \in Λ) \in V_{η} \cap V_{μ}

, it is necessary that there be no element in the intersection itself. If there were any element

{[p]}_{\sim} \in V_{η} \cap V_{μ}

, surjectivity guarantees that there necessarily exists some

p \in Λ

. Therefore:

\forall {Preim}_{q} (V_{η} \cap V_{μ}) = \emptyset \Rightarrow V_{η} \cap V_{μ} = \emptyset .

Let the topological space

(Λ / \sim, {\tilde{T}}_{Λ})

\forall {[p_{μ}]}_{\sim}, {[p_{η}]}_{\sim} \in Λ / \sim, {[p_{μ}]}_{\sim} \neq {[p_{η}]}_{\sim}, \exists V_{μ} \in {\tilde{T}}_{Λ} \land V_{η} \in {\tilde{T}}_{Λ} : V_{η} \cap V_{μ} = \emptyset .

Therefore, the treon quotient topology

Λ / \sim

is a Hausdorff space.

All this analysis allows us to well-define the treon quotient topology

Λ / \sim

as a Hausdorff space, without the need to equip the vector space with norms or inner products. The definition of the product in Bermejo Algebras is sufficient to work with these concepts implicitly.

We must consider that we can also define a canonical projection mapping

m : Λ \to Λ / \sim

that is not given by the composition

q \circ H

, and therefore does not depend on conventional vectors

\vec{p}

; however, this excludes from our analysis the equivalence classes of opposite vectors in the

S_{i}^{2}

r-spheres and returns surface areas of radius

r_{i}

. Let us consider the mapping m:

m : Λ \to Λ / \sim

p \mapsto {[p]}_{\sim}

where:

\forall p_{i}, p_{j} \in Λ : p_{i} \sim p_{j} \Leftrightarrow \sqrt{Re (〈 p_{i}^{2} 〉)} = \sqrt{Re (〈 p_{j}^{2} 〉)} .

Note that now the equivalence classes are not two points represented by position vectors on

S_{i}^{2}

but are each of the surfaces

S_{i}^{2}

Similarly, we can now define other equivalence classes in their corresponding

Λ / \sim

, for example, taking the diamond ⋄ operation:

\forall p_{i}, p_{j}, p_{k}, p_{l} \in Λ : (p_{i}, p_{j}) \sim (p_{k}, p_{l}) \Leftrightarrow p_{i} ⋄ p_{j} = p_{k} ⋄ p_{l} .

Our work thus opens new possibilities in the development and topological analysis of Bermejo Algebras.

4. Conclusions

We conducted a topological analysis of the Bermejo Algebras, demonstrating their ability to create Hausdorff spaces without the need for traditional norms, inner products, or metrics. We achieved this distinctive feature through the creation of equivalence classes and a specific quotient topology. This contribution is particularly relevant for the study of complex functions and complex analysis in a non-associative algebraic framework.

We defined a Bermejian inner product derived from the product of the Bermejo Algebras; this product is distinguished by not depending on conventional definitions where vector spaces are equipped with metric, inner product, and norm operations. This Bermejian inner product naturally arose from the product of the Bermejo Algebras, allowing us to work with metric and norm properties without explicitly introducing these structures.

Our work established an important precedent for the exploration of new algebraic structures, topological structures, and differential varieties. By constructing treonic spheres and treonic equivalence classes, we provided a new way to group and analyze treons. Demonstrating that the treonic quotient space is a Hausdorff space ensured the uniqueness of the limits of point sequences in these spaces, which is fundamental for the definition and study of differential varieties.

This work set an important precedent for the exploration of new algebraic and topological structures, opening a field of research for mathematicians and physicists.

References

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Hausdorff Spaces in Bermejo Algebras: The Birth of Treonic Manifold Construction

Abstract

1. Introduction

2. Bermejo Algebras

3. Construction of a Topology on Treons

3.1. Definitions for Treonic Topology

3.2. Convergence of Treons and Definition of Hausdorff Space

3.3. Definition of Treonic Quotient Topology

3.4. Analysis of the Real Component of p ⊙ p ( ( * i , j ) )

3.5. S r 2 -Spheres and Treonic Equivalence Classes

4. Conclusions

References

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3.4. Analysis of the Real Component of $p ⊙ p^{((*_{i, j}))}$

3.5. $S_{r}^{2}$ -Spheres and Treonic Equivalence Classes