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Universidad Técnica de Manabí
DOI: https://doi.org/10.33936/revbasdelaciencia.v7iESPECIAL.4722
UN ESTUDIO EXPLORATORIO SOBRE TOPOLOGÍAS PRIMALES
Carlos Garcia-Mendoza
1*
, Jorge Enrique Vielma
2
, José Játem
3
1
Estudiante Maestría Académica con Trayectoria de Investigación en Matemática. Instituto de Ciencias Básicas.
Universidad Técnica de Manabí. Ecuador. Correo electrónico: daniel.garcia@utm.edu.ec
2
Facultad de Ciencias Naturales y Matemáticas. Escuela Superior Politécnica del Litoral. Ecuador. E-mail:
jevielma@espol.edu.ec
3
Coordinación de Matemáticas. Universidad Simón Bolívar. Venezuela. E-mail: jrjatem@gmail.com
*Autor para la correspondencia: daniel.garcia@utm.edu.ec
Recibido: 26-05-2022/ Aceptado: 12-12-2022 / Publicación: 26-12-2022
Editor Académico: Luis Bladismir Ruiz Leal
RESUMEN
Sea un conjunto no vacío y una función. La colección
es llamada topología
primal inducida por sobre . Con esta topología, el espacio es un espacio Alexandroff, es decir, la intersección de
una familia arbitraria de abiertos es un conjunto abierto. Una topología provista de dos operaciones biarias definidas a
través de la unión e intersección de conjuntos puede ser vista como un semianillo. El objetivo de este trabajo es mostrar
algunas de las propiedades generales de las topologías primales, en particular, las características de las topologías primales
vistas como semianillos. Entre otros resultados, proveemos algunos de los ideales primos y maximales que se pueden
contruir para una topología primal arbitraria. Finalmente, proveemos algunos resultados relacionados a las topologías
primales inducidas por un homomorfismo de grupos sobre un grupo .
Palabras clave: Espacio primal, semianillo, homomorfismo de grupo.
AN EXPLORATORY STUDY ON PRIMAL TOPOLOGIES
ABSTRACT
Let be a non-empty set and a function. The collection
is called primal topology
induced by on . With this topology, the space is an Alexandroff space, that is, the intersection of an arbitrary family
of open sets is an open set. A topology equipped with two binary operations defined through the union and intersection
of sets can be seen as a semiring. The aim of this paper is to show some of the general properties of primal topologies, in
particular, the characteristics of primal topologies seen as semirings. Among other results, we provide some of the prime
and maximal ideals that can be constructed for an arbitrary primal topology. We finally provide some results related to
the primal topology induced by a group homomorphism on a group .
Keywords: Primal space, semiring, group homomorphism.
Ciencias Matemáticas
Carlos Garcia-Mendoza, Jorge Enrique Vielma, José Játem
108
UM ESTUDO EXPLORATÓRIO SOBRE TOPOLOGIAS PRIMAIS
RESUMO
Seja um conjunto não vazio e uma função. A coleção
é chamada de topologia
primal induzida por em . Com esta topologia, o espaço é um espaço de Alexandroff, ou seja, a interseção de uma
família arbitrária de conjuntos abertos é um conjunto aberto. Uma topologia provida de duas operações binárias
definidas por união e interseção de conjuntos pode ser vista como um semi-anel. O objetivo deste artigo é mostrar algumas
das propriedades gerais das topologias primais, em particular, as características das topologias primais vistas como semi-
anéis. Entre outros resultados, fornecemos alguns dos ideais primos e máximos que podem ser construídos para uma
topologia primal arbitrária. Finalmente, fornecemos alguns resultados relacionados à topologia primal induzida por um
homomorfismo de grupo em um grupo .
Palavras chave: espaço primal, semi-anel, homomorfismo de grupo
Citación sugerida: Garcia-Mendoza, C., Vielma, J., Lasser, J. (2022). Un estudio exploratorio sobre topologías primales.
Revista Bases de la Ciencia, 7 (No Especial), Diciembre, 107-121. DOI:
https://doi.org/10.33936/revbasdelaciencia.v7i3.4722
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1. INTRODUCTION
Alexandroff (1937) introduced the notion of topological spaces in which the arbitrary union of closed
sets is a closed set, which he named Diskrete Raume or Discrete Spaces. This property is clearly
equivalent to the fact that the arbitrary intersection of open sets is an open set. A trivial example of
these spaces, on which mathematicians such as McCord (1966), Stong (1966) and Herman (1990)
worked on, correspond to finite topological spaces. However, given that the term discrete space was
already being used to describe those on which every subset is open, McCord, for instance, decided to
rename them as A-spaces, and focused his study on
A-spaces. Herman (1990) on the other hand,
named these spaces Sparse, but mathematicians would eventually opt for the name Alexandroff
spaces in honor of Pavel Alexandroff, whom initially drew the attention to this topic.
Shirazi and Golestani (2011) would later present a study on a proper subclass of Alexandroff spaces,
which they called Functional Alexandroff Spaces, given that the topology considered was induced by
a function, and the resultant space was indeed Alexandroff. Finally, Echi (2012) presented some
results about these spaces and named them Primal spaces, term that has been used since. In this paper,
we equip a primal topology
with two operations defined through the union and intersections of
sets, which make
a semiring. We present some of the properties of these topologies seen as
semirings, more specifically, we show some of the prime and maximal ideals that can be constructed
for a primal topology
. We also study the generalization of some results previously presented by
the authors of this paper. In particular, we show the generalization of primal spaces induced on a
finite dimension vector space by the use of group homomorphisms.
2. PRELIMINARIES
In this section we present some of the fundamental concepts about primal topologies.
Definition 2.1. Given a non-empty set and a function , then the collection
is called primal topology induced on , and the space
is called primal
space.
Equivalently, a primal topology
on a set can be defined by deciding the closed sets to be those
subsets that are - invariant, that is,
. In order to see this equivalence, let be an
open set of a primal space
, then by definition is closed. If we assume that the image of
under is not contained in , then there exists an element such that
Carlos Garcia-Mendoza, Jorge Enrique Vielma, José Játem
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, a contradiction. Some of the well-known properties of the function associated to a primal space
are given below.
Lemma 2.1. The function associated to the primal space
is continuous.
Proof: Let be an open set of . In order for
to be open, it must hold
. Let
. Since A is open, we have
, which implies . By the same
hypothesis we then have
.
The following two Lemmas are provided by Shirazi and Golestani (2011), for which we provide the
proofs.
Lemma 2.2. Let
be a primal space. Then is a homeomorphism if and only if it is a
bijection.
Proof: If is a homeomorphism, it is trivial that is a bijection. For the converse statement, by
Lemma 2.1 we have that is continuous. In order for
to be continuous, it must hold
, for
. Given that in injective, then
, which implies that
must
hold. This is true given that
.
Lemma 2.3. Let
be a primal space. Then is a homeomorphism if and only if it is
injective and open.
Proof: If is a homeomorphism, it is trivial that it is an injective and open function. For the converse
statement, by Lemma 2.1 we have that is continuous. For
to be continuous, it must hold that
, for
, which is true given that is open.
We shall now prove that primal topologies are indeed Alexandroff topologies. In order to do that, we
first define Alexandroff spaces and provide one of its characterizations.
Definition 2.2. Let be a topological space, then is an Alexandroff space if the arbitrary
intersection of open sets is an open set.
The following theorem, which characterizes Alexandroff spaces, is given by Speer (2007).
Theorem 2.1. is an Alexandroff space if and only if every point has a minimal open
neighborhood.
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We resort to this result in order to prove that every primal space is Alexandroff. For this reason, we
define a set for every point in a primal space which will be proven to be the minimal open
neighborhood of the point.
Definition 2.3. Let
be a primal space and . A preorder
can be defined on as
follows:
if and only if there exists an integer such that
.
Definition 2.4. Let
be a primal space. Then the sets:
are called the closure and kernel of respectively.
Proposition 2.1. Let
be a primal space and , then is the minimal open set
containing .
Proof: We assume there exists an open set
such that . Then there exists an
element
. Moreover, there exists an integer such that
and
, which implies
, a contradiction.
Corollary 2.1. Every primal space is an Alexandroff space.
Proof: It is an immediate result from Theorem 2.1 and Proposition 2.1.
We now show some of the fundamental properties of the sets defined above.
Lemma 2.4. Let be two distinct elements of a primal space
. Then
or
or
.
Proof: We assume
. Then there exists an element and integers
such that
and
. Given that we have that . If we assume
then and
. If we assume then
and
Lemma 2.5. Let
be a primal space. Then for all we have is the minimal closed
set containing .
Proof: We assume there exists a closed set of such that
. Then there exists an
element
and an integer such that
and
, which implies
, a contradiction.
Carlos Garcia-Mendoza, Jorge Enrique Vielma, José Játem
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Therefore, any closed set of can be expressed as a union of the closure of each element of .
More precisely, if is a closed set of then
. Observe that given that every primal
space is Alexandroff (equivalent to the fact that the arbitrary union of closed sets is a closed set), we
have that
is indeed a closed set.
Lemma 2.6. Let
be a primal space and an open subset of , then is a primal space
equipped with the subspace topology:
Proof: Let be an element of
, then for some
. Also
and given that and are open set of it follows
. Therefore
and
is a primal topology.
In regards of the connected components of a primal space, Shirazi and Golestani (2011) show the
following result.
Lemma 2.7. Let
be a primal space, then any two elements are in the same connected
component if and only if there exist such that
.
Given this Lemma, it is possible to define an equivalence relation
for
as follows: for any
two elements ,
if and only if there exist integers such that
.
Proposition 2.2. The relation
is indeed an equivalence relation.
Proof: Reflexivity and symmetry are trivial. To prove transitivity, let
and
, then there
exist integers such that
and
. Therefore
, and
.
Naturally, it is possible to define the equivalence class of each element as the set
and the quotient space as the set
3. GENERAL RESULTS ABOUT PRIMAL SPACES
In this section we present some general results about primal spaces. In particular, we focus on some
of the characteristics of connected components in primal spaces.
Lemma 3.1. Let
be a primal space and , then
.
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Proof: Let
, then there exists an integer such that
. Given that then
there exist integers such that
Applying the result from Proposition 2.2 we
have
which implies
and .
Given this result, it is possible to show that every connected component of a primal space is closed.
This result is also shown by Garcia-Mendoza et al (2021), however we provide a different approach
to prove such property.
Lemma 3.2. Let
be a primal space, then
.
Proof: Let , since and
we have
. On the other
hand, let
, then there exists such that and by Lemma 3.1 we have
, which implies
.
Note that every connected component is clopen, since the equivalence classes form a partition of the
space , and the complement of each connected component is the union of closed connected
components. The following result is shown by Guale et al (2020).
Lemma 3.3. Let
be a connected primal space and let
. If , then or
.
We propose the following generalization of the previous Lemma as follows.
Lemma 3.4. Let
be a connected primal space such that
is finite for every . Let
be a collection of open sets of such that
. Then there exists
such that
.
Proof: If is a connected primal space and is finite for all then there exists such
that is a periodic point. It must hold that
for some
. It also holds that
, which implies
.
Observe that this result cannot be extended for the case where is infinite for all is a connected
primal space. In order to see this, it is necessary to introduce the following equivalence.
Lemma 3.5. Let
be a connected primal space, then the following are equivalent:
a. If
for open sets
of , then there exists such that
b. There exists such that
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Proof: : Consider the following union
which is clearly equal to . From a. it
follows that there exists such that
. : Consider a collection of open sets
with such that
By b. it follows that there exists such that
. By
Proposition 2.1 we have that
for some .
Observe that this equivalence implies the existence of a periodic point . To prove this, assume
proposition b. from Lemma 3.5 as true. Let such that
then is also an element
of and by b. it is also an element of , which implies that there exists an integer such
that
, therefore is a periodic point. The existence of a periodic point
impedes the existence of an infinite for some in a connected primal space . In order to
illustrate this, consider the following trivial example.
Example 3.1. Let be the set of natural numbers and a function defined by
.
Then
is a connected primal space and is infinite for all .
If we assume there exists a collection of open sets
such that
, then by the
equivalence shown in Lemma 3.5 it must hold that there exists such that
, a contradiction.
4. PRIMAL TOPOLOGIES SEEN AS SEMIRINGS
In this section we study some of the properties of primal topologies seen as semirings. In particular,
we show some of the prime ideals that can be constructed for an arbitrary primal topology.
Definition 4.1. A semiring is a set equipped with two binary operations and , called addition
and multiplication respectively, such that:
•
is a commutative monoid with identity element
•
is a monoid with identity element
• Multiplication distributes over addition
• Multiplication by annihilates .
Example 4.1. The set of all square matrices with positive entries with the usual addition and
multiplication between matrices is a semiring.
Lemma 4.1. If is a topological space, then is a semiring with and
as the addition and multiplication operations.
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Proof: The union and intersection of sets are clearly binary operations. Moreover, the identity element
of the addition and multiplication operations are and respectively. The commutativity,
associativity and distributity of these operations are obtained from the properties of union and
intersection of sets. Finally, it is clear that multiplication by annihilates .
Given that the multiplication operation is defined through the intersection of sets, a commutative
binary operation, we have that a topology is actually a commutative semiring, equipped with the
operations above defined.
Definition 4.2. A subset of a semiring
is called an ideal of , if the identity element of the
addition operation is an element of , and for every and it holds .
Definition 4.3. A semiring homomorphism from a semiring to a semiring is a function
such that for all it holds:
5.
5.
5.
Lemma 4.2. Let be a semiring homomorphism and let be an ideal of , then
is an ideal of S.
Proof: It is easy to see that
. Let
, then it holds
, and given
that is an ideal, then
. Given that is a semiring homomorphism, then
, which implies
Using similar arguments it can be seen that
and
is an ideal of .
Definition 4.4. A proper ideal of a semiring
is called a prime ideal of , if implies
or .
Definition 4.5. A proper ideal M of a semiring
is called maximal ideal of if , for
an ideal of , then or .
Note that for an ideal to be a proper ideal of a semiring it must hold that , which in turn
requires .
Lemma 4.3. Let
be a topological space, then
is a prime ideal of .
Proof: It easy to see that
. Let and . Given that and then
which implies
. It is trivial that
. If for
Carlos Garcia-Mendoza, Jorge Enrique Vielma, José Játem
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then or . Indeed, if we assume and then and
, a contradiction. It must hold then that or , that is or .
Theorem 4.1. If
is a
topological space, then
is a maximal ideal of
.
Proof: Given that is
, then is open, moreover
. If we assume is
not a maximal ideal of , then there exists an ideal of such that
. Therefore, there
exists an open set such that . Then
, which implies . It follows
that is a maximal ideal of .
Theorem 4.2. Let
be a primal space and a connected component of . Then
is a prime ideal of
.
Proof: It is easy to see that
. Let ,
and a connected component of .
If
, then applying Lemma 2.6 and Lemma 3.3 we have or , a contradiction.
Therefore and
. It is trivial that
. Finally, if
for
, then or . Indeed, if we assume then , and
given that , it must hold that , that is
. A similar result is obtained if we
assume
Garcia-Mendoza et al (2021) further showed that this ideal is also a maximal ideal of
. Note that
the axioms for an ideal of a semiring only consider the addition of a finite number of elements of the
ideal to be an element of the ideal itself. However, it is also possible to show that for a primal space
with a connected component composed of finite orbits, that the arbitrary union of elements of the
ideal is also an element of the ideal. For that, consider the collection
. If we
assume
then applying Lemma 2.6 and Lemma 3.4 we have that there exists an open
set
from the given collection such that
, a contradiction. It must hold then that
, and therefore
.
Theorem 4.3. Let
be a primal space and the orbit of a point, then
is a prime ideal of
.
Proof: Let
and
. Since and then
, which
implies
. It is trivial to see that , moreover given that
. It also holds that if for some
then or
. Indeed,
let and , if then
, a contradiction. On the other
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hand, and without loss of generality, we assume and , then contains and
. Therefore which implies
, a contradiction. It follows that
or .
Theorem 4.4. Let
be a primal space and a subset of a periodic orbit , then
is a prime ideal of
.
Proof: Let and
. Since and then
, which
implies . It is clear that , moreover since . Finally,
if then or . Indeed, if we assume and
where are two distinct points of , then we have that
, and given that is a
periodic orbit, then . Therefore , a contradiction. It follows that or
.
As with the previous ideal, it is possible to show that the arbitrary union of elements of the ideal
is also an element of the ideal itself.
Proposition 4.1. Let
be a primal space, a subset of and
,
then
.
Proof: If we assume
, then
, therefore there exists such
that
and . Then it must hold that
for some
. This implies
, a contradiction.
We now present some results regarding the primal topology induced on
by a linear transformation
. Recall that any linear transformation can be written in matrix form, that is, if is a
linear transformation, then it can be written as
with a square matrix and
.
For instance, the linear transformation
associated with vector rotation in
can be
written as follows:
With
and . This linear transformation induces a primal topology on
, and it can be
seen that regardless of the choice of , the primal space
will have an infinite number of
connected components. This makes
a non-connected topological space, characteristic that is not
obtained when
is equipped with the usual topology. Additionally, this transformation is invertible,
Carlos Garcia-Mendoza, Jorge Enrique Vielma, José Játem
118
given that the associated matrix is invertible. Other interesting results arise from considering linear
transformations associated with real entries matrices as the following:
Example 4.2. Consider the following diagonal matrix
The function
defined as with
induces a primal topology on
, which
can be denoted by
.
It is evident that the open sets of this space will strongly depend on the values of the diagonal. If, for
instance, one of the diagonal values
, then
would be an infinite set given that every
vector
with all coordinates equal to 0 except for the i-th coordinate will satisfy the equality
. Note as well that in this case the matrix will not be invertible, and so the linear
transformation is not invertible. On the other hand, if all values on the diagonal are distinct from zero,
the determinant of the matrix A is not zero and the matrix is invertible. In this case the following
holds.
Lemma 4.4. Let
be a diagonal matrix. If every
then
and
.
Proof: If every
, then for the system
we have
, then
for
which implies
. Therefore
and
.
In general, the following result is obtained:
Corollary 4.1. Let
be a diagonal matrix. If every
then
is
clopen.
Note as well that
equipped with this topology is not a connected topological space. We may now
consider arbitrary linear transformations associated with a matrix of real entries.
Lemma 4.5. If is an invertible matrix and
, then
.
Proof: We have
. But since
i.e.,
, then
which implies
.
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Lemma 4.6. If is an invertible matrix, then
is an ideal of
.
Proof:
, then . Let , then
, therefore
. Moreover,
, given that
and
. If and
, then
. Moreover
since
, then
Theorem 4.5. If is an invertible matrix, then
is a maximal ideal of
.
Proof: If is an ideal of
such that then there exists such that
. In particular
we have that
. By Lemma 4.5 we have that
and
. Therefore
, then
and is
maximal.
5. GROUP HOMOMORPHISMS AND PRIMAL TOPOLOGIES
In this section we provide some generalizations to some results shown by Garcia-Mendoza et al
(2021). More specifically, we generalize the notion of primal topologies induced on finite dimension
vector spaces.
Theorem 5.1. If the function that defines the primal space
is the identity function,
then the space is zero-dimensional.
Proof: If is the identity function, then
is the discrete topology and for each we have
. From Lemma 3.3 by Garcia-Mendoza et al (2021), we have
is a maximal ideal.
In order to prove this Lemma, it is enough to show that any given prime ideal is also a maximal ideal.
We prove this by showing that for a given prime ideal of
there exists such that and
. Let
, then
. Given that
then it must hold and
. Since
is maximal, then
.
Definition 5.1. Let be a group homomorphism. The set
is
called the algebraic kernel of , where
is the identity element of the group
It is a well-known result that if is a group homomorphism then is 1-1 if and only if
. We provide a topological characterization of 1-1 group homomorphisms.
Theorem 5.2. Let be a group and a group homomorphism. Then is 1-1 if and only if
.
Carlos Garcia-Mendoza, Jorge Enrique Vielma, José Játem
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Proof: Given that is a group isomorphism we have
, that is,
, which means
and
. On the other hand, if
then
given that
, then
, which implies
and
is 1-1.
Corollary 5.1. If is a group anf is an injective group homomorphism then is clopen.
Proof: By the previous Theorem we have that is open. Given that
then
is also
closed.
Theorem 5.3. Let be a group homomorphism and a prime ideal of
. If
then
is maximal and .
Proof: Given that is a group homomorphism then
, which is equivalent to
, and
by Lemma 3.3 by Garcia-Mendoza et al (2021) we have is a maximal ideal of
. Let ,
then , which implies
. Given that is a prime ideal of
and
then
. Therefore
and given that is maximal we have
.
6. CONCLUDING REMARKS
In this paper we explored some of the properties of primal topologies seen as semirings. Some of the
advantages of studying the algebraic properties of these topologies is reflected, for example, in
Theorem 4.5. where it was possible to construct another algebraic condition for the invertibility of a
matrix considering the topological and algebraic properties of the primal topology induced on
by
the matrix. Moreover, considering topologies as semirings has opened the door to address certain
problems in a novel way. For instance, the Collatz conjecture, a problem that has not been solved yet,
can be studied from a topological point of view, given that the topology induced on is a primal
topology. In this paper, some algebraic structures such as ideals, maximal and prime ideals are
considered, which could shed light on the study of the conjecture.
7. DISCLOSURE OF CONFLICT OF INTEREST OF THE AUTHORS
The authors declare no conflict of interest.
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Authors’ contribution
Author
Contribution
Carlos Garcia-Mendoza
Writing of manuscript, bibliographic search, proofs
Jorge Enrique Vielma
Proposal of main theorems, manuscript structure
José Játem
Proposal of generalizing theorems, manuscript proof reading
