Divulgaciones
Matemáticas
p-ISSN 1315-2068
Depósito legal: pp 199302ZU392
Maracaibo - Venezuela
Departamento de Matemática
Vol. 23-24 - No. 1-2 - 2022-2023
Facultad
Experimental
de Ciencias
Universidad
del Zulia
e-ISSN 2731-2437
Divulgaciones Matemáticas
Revista Matemática de la Universidad del Zulia
Facultad Experimental de Ciencias
Departamento de Matemática
Revista arbitrada, publicada de forma digital, de libre acceso, indizada en Latindex, Wordcat,
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@
bel, MIAR, Dialnet, EuDML, Mathematical Reviews, MathSci online/CD-ROM, Zentral-
blatt für Mathematik, Revencyt y REDIB. Se publica un volumen anual compuesto por dos
números, que aparecen en junio y diciembre.
Comité Editorial
Dr. Tobías Rosas Soto (LUZ)
Dr. Vinicio Ríos (LUZ) Dr. Wilson Pacheco (LUZ)
Dr. Deivi Luzardo (LUZ)
Editor Jefe:
Dr. Tobías Rosas Soto (
trosas@demat-fecluz.org
)
Editores Asociados:
Dr. Vinicio Ríos, Dr. Wilson Pacheco
Editores Eméritos:
Dr. Alirio J. Peña P., MSc. Ángel V. Oneto R., Dr. José H. Nieto S., Dr.
Genaro González, Dr. Daniel Núñez.
Editore Fundadores:
Dr. Alirio J. Peña P., MSc. Ángel V. Oneto R.
Portada diseñada por Tobías Rosas Soto. Dirección Postal:
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Depósito Legal pp 199302ZU392
p-ISSN: 1315-2068
Depósito Legal pe ZU2021000035
e-ISSN: 2731-2437
Compuesta con L
A
T
E
X y
A
M
S
-L
A
T
E
X en el Departamento de Matemática de la Facultad Experi-
mental de ciencias, Universidad del Zulia.
c
1993 La Universidad del Zulia.
Universidad del Zulia
Maracaibo, Venezuela
DIVULGACIONES
MATEM
´
ATICAS
Vol. 23-24
2022-2023
No. 1-2
Presentación
El Comité Editorial de
Divulgaciones Matemáticas
se complace en presentar el
Vol. 23-
24
,
No. 1-2
,
2022-2023
. En el presente volumen se resumen todos los artículos recibidos entre
los años
2022
y
2023
, los mismos fueron evaluados y aceptados para su publicación. Esta edición
nace como una decisión del Comité Editorial en busca de revivir las actividades de la revista, las
cuales se han visto menguadas por la poca demanda de artículos por parte de los autores por
diversas razones. Entre éstas gura el hecho de que la revista no aparece reejada en SCOPUS,
por la no continuidad de publicación de números y volúmenes de la revista, por la falta de tra-
bajos sometidos a la revista paradójicamente.
Los trabajos publicados en esta edición mostrarán la fecha en la que se recibieron y la fecha
en la que fueron aprobados. Todos los trabajos recibidos en los años 2022 y 2023 fueron artículos
de investigación, dichos trabajos se distribuyen de la siguiente manera: cuatro (4) artículos en
el año 2022, y cinco (5) artículos en el año 2023. En cada año solo un artículo, de los recibidos,
no aprobó la etapa de evaluación por los árbitros respectivos. De manera que en este número se
publican solo siete (7) artículos en la sección de Artículos de Investigación. Por otro lado, en la
sección de Problemas y Soluciones, se proponen dos (2) problemas, y se presenta la solución del
problema No. 27 propuesto en Vol. 8, No. 2, 2000 de la revista.
El trabajo editorial relacionado con este número es el resultado de mucho esfuerzo del Comité
Editorial y del Editor Jefe de la revista. Los Editores queremos expresar nuestro agradecimiento
a todos aquellos que hicieron posible este volumen: a los autores de los trabajos que se presentan,
que dieron su voto de conanza a la revista; a los árbitros que evaluaron los artículos, cuya labor
desinteresada permitió satisfacer los estándares de calidad de la revista y mejorar sensiblemente la
forma de los trabajos; al equipo editorial de
Divulgaciones Matemáticas
. A todos, mil gracias.
La revista está ahora en el portal de
Revistas Cientícas y Humanísticas de la Univer-
sidad del Zulia (ReviCyHLUZ)
cuyo sitio web ocial es:
produccioncientificaluz.org
.
Ahora los artículos están identicados con el membrete del
Sistema de Servicios Bibliote-
carios y de Información de LUZ (SERBILUZ)
, y la revista pasa a tener como sitio web
ocial
produccioncientificaluz.org/index.php/divulgaciones
.
Es importante aclarar que la dirección web
divmat.demat-fecluz.org
continúa funcionando
para obtener los números de la revista publicados antes del año 2016, hasta que los mismos sean
trasladados en su totalidad al nuevo sitio web mencionado. Todo esto con la nalidad de darle
más expansión y reconocimiento a la revista.
Por último, el Comité Editorial de
Divulgaciones Matemáticas
pide disculpas a los autores
de los artículos aquí publicados por el notable retraso en la publicación de este número y por los
inconvenientes que esto pudo haberles causados, les agradecemos su espera. Además, invitamos
a la comunidad matemática venezolana e internacional a seguir dándonos su voto de conanza
sometiendo sus trabajos en la revista para evaluación y posible publicación.
1
Dr. Tobías Rosas Soto.
1
Editor en Jefe de
Divulgaciones Matemáticas
y editor del presente número
Presentation
The Editorial Committee of
Divulgaciones Matemáticas
is pleased to present
Vol. 23-
24
,
No. 1-2
,
2022-2023
. This volume brings together all the articles received between the years
2022
and
2023
, they were evaluated and accepted for publication. This edition was born as a
decision of the Editorial Committee in search of reviving the activities of the journal, which have
been diminished by the low demand for articles by the authors for various reasons. Among these
is the fact that the journal is not reected in SCOPUS, due to the non-continuity of publication of
issues and volumes of the journal, due to the lack of works submitted to the journal, paradoxically.
The works published in this edition will show the date they were received and the date they
were approved. All the works received in the years 2022 and 2023 were research articles, these
works are distributed as follows: four (4) articles in the year 2022, and ve (5) articles in the
year 2023. In each year only one article, of those received, did not pass the evaluation stage
by the respective referees. So in this issue only seven (7) articles are published in the Research
Articles section. On the other hand, in the Problems and Solutions section, two (2) problems are
proposed, and the solution to problem No. 27 proposed in Vol. 8, No. 2, 2000 of the magazine
is presented.
The editorial work related to this issue is the result of much eort by the Editorial Committee
and the Editor-in-Chief of the journal. The Editors would like to express our gratitude to all
those who made this volume possible: to the authors of the works presented, who gave their vote
of condence to the journal; to the referees who evaluated the articles, whose seless work made
it possible to satisfy the quality standards of the journal and signicantly improve the form of
the works; to the editorial team of
Divulgaciones Matemáticas
. Thank you all.
The journal is now on the portal of
Scientic and Humanistic Magazines of the
University of Zulia (ReviCyHLUZ)
whose ocial website is:
produccioncientificaluz.
org
. Now the articles are identied with the letterhead of the
LUZ Library and Infor-
mation Services System (SERBILUZ)
, and the journal now has as its ocial website
produccioncientificaluz.org/index.php/divulgaciones
.
It is important to clarify that the web address
divmat.demat-fecluz.org
continues to func-
tion to obtain the issues of the journal published before 2016, until they are transferred in their
entirety to the new website mentioned. All this with the purpose of giving more expansion and
recognition to the journal.
Finally, the Editorial Committee of
Divulgaciones Matemáticas
apologizes to the authors
of the articles published here for the notable delay in the publication of this issue and for the
inconveniences that this may have caused them, we thank them for their wait. Furthermore, we
invite the Venezuelan and international mathematical community to continue giving us their vote
of condence by submitting their work to the journal for evaluation and possible publication.
2
Dr. Tobías Rosas Soto.
2
Chief Editor of
Divulgaciones Matemáticas
and editor of the present volume
DIVULGACIONES MATEMÁTICAS
Vol. 23-24, No. 1-2, 2022-2023
Contenido
(Contents)
:
Artículos de Investigación
(Research papers)
Super quasi-topological and paratopological vector spaces versus topolog-
ical vector spaces.
Super casi-topológicos y paratopológicos espacios vectoriales versus espacios vectori-
ales topológicos.
Madhu Ram, Bijan Davvaz
111
Los números Ramsey para tres grafos y tres colores.
The Ramsey numbers for three graphs and three colors.
José Figueroa, Tobías Rosas, Henry Ramírez, Armando Anselmi
1228
Estudio cualitativo del metabolismo de una droga ingerida.
Qualitative study of the metabolism of an ingested drug.
Berónica Aguilar - Adolfo Fernández - Sandy Sánchez - Antonio Ruiz
2943
T op(X)
y
Spec(τ )
como espacios primales.
T op(X)
and
Spec(τ )
as primal spaces.
Viviana Benavides - Jorge Vielma
4453
Grafo divisor de cero de
Z
2
r
q
s
.
Zero divisor graph of
Z
2
r
q
s
.
Juan Otero - Daniel Brito - Tobías Rosas
5463
Un método nuevo para eliminar la indeterminación en los problemas sin-
gularmente perturbados con resonancia de Ackerberg y O'Malley.
A new method for eliminating the indeterminacy in the singularly perturbed problems
with Ackerberg-O'Malley resonance.
Jacques Laforgue
6481
Boundary Estimation with the Fuzzy Set Regression Estimator.
Estimación Frontera con el Estimador de Regresión con Conjunto Difuso.
Jesús Fajardo
82106
Problemas y Soluciones
(Problems and Solutions)
Tobías Rosas Soto.
(Editor) 107112
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
https://produccioncientificaluz.org/index.php/divulgaciones/
DOI: https://doi.org/10.5281/zenodo.11515886
(CC BY-NC-SA 4.0)
c
Autor(es)
e-ISSN 2731-2437
p-ISSN 1315-2068
Super quasi-topological and paratopological
vector spaces versus topological vector spaces
Super casi-topol´ogicos y paratopol´ogicos espacios vectoriales versus espacios
vectoriales topol´ogicos
Madhu Ram (madhuram0502@gmail.com)
ORCID: https://orcid.org/0000-0001-6583-0978
Department of Mathematics
University of Jammu
Jammu-180006, Jammu & Kashmir, India.
Bijan Davvaz (davvaz@yazd.ac.ir)
Department of Mathematics
Yazd University
Iran.
Abstract
In this paper, we introduce the idea of super quasi-topological vector space which is
an extension of the concept of topological vector space and investigate some of its basic
properties. We extend the existing notion of quasi-topological vector space to all complex
vector spaces and investigate the relationship of super quasi-topological vector spaces with
paratopological and quasi-topological vector spaces.
Palabras y frases clave: Topological vector space, paratopological vector space, quasi-
topological vector space, super quasi-topological vector space, quotient space.
Resumen
En este art´ıculo, presentamos la idea del espacio vectorial supercuasitopol´ogico, que es
una extensi´on del concepto de espacio vectorial topol´ogico, e investigamos algunas de sus
propiedades b´asicas. Extendemos la noci´on existente de espacio vectorial cuasi-topol´ogico
a todos los espacios vectoriales complejos e investigamos la relaci´on de los espacios vecto-
riales s´uper cuasi-topol´ogicos con los espacios vectoriales paratopol´ogicos y cuasi-topol´ogicos.
Key words and phrases: Espacio vectorial topol´ogico, espacio vectorial paratopol´ogico,
espacio vectorial cuasi-topol´ogico, espacio vectorial supercuasi-topol´ogico, espacio cociente.
1 Introduction
Recall that a paratopological group is a group G with a topology such that the group operation
of G is continuous. If in addition, the inversion map in a paratopological group is continuous,
then it is called a topological group.
Recibido 11/03/2022. Revisado 7/04/2022. Aceptado 21/09/2022.
MSC (2010): Primary 57N17; Secondary , 57N99.
Autor de correspondencia: Madhu Ram
2 Madhu Ram - Bijan Davvaz
According to [2], a real vector space L endowed with a topology τ such that (L, +, τ ) is a
paratopological group, is called:
(1) paratopological vector space if for each neighborhood U of λx with x ∈ L and λ ∈ R
+
(the
set of non-negative real numbers), there exist a neighborhood V of x and an > 0 such
that [λ, λ + [.V ⊆ U.
(2) quasi-topological vector space if the function H
r
: L → L defined by H
r
(x) = rx with
r ∈ R
+
, is continuous.
Hence, all translations and dilations of a paratopological (resp. quasi-topological) vector
space are homeomorphisms. For more details, see [1] and [2]. Paratopological vector spaces were
discussed and many results have been obtained (for example, see [1], [2], [3] and [4]).
Lemma 1.1. (cf. [2]) For a real vector space L with a topology τ, the following conditions are
equivalent.
I. (L, τ ) is a paratopological vector space.
II. There exists a local basis B at 0 of L satisfying the following conditions:
(a) for every U, V ∈ B, there exists W ∈ B such that W ⊆ U ∩ V ;
(b) for each U ∈ B, there exists V ∈ B such that V + V ⊆ U ;
(c) for each U ∈ B and for each x ∈ U, there exists V ∈ B such that x + V ⊆ U;
(d) for each U ∈ B and for each r > 0, rU ∈ B;
(e) each U ∈ B is absorbent and quasi-balanced.
Motivated by the papers [2] and [3], the aim of this paper is to introduce and study the
super quasi-topological vector spaces. Relationship of super quasi-topological vector spaces with
paratopological, quasi-topological and topological vector spaces is investigated.
In the following, all vector spaces are over the field F ∈ {R, C}. For any undefined concepts
and terminologies, refer to [8].
2 Relationship among various classes of topological vector
spaces
In this section, we define super quasi-topological vector space and extend the definition of
paratopological and quasi-topological vector space to all complex vector spaces. Then we in-
vestigate the relation between super quasi-topological, quasi-topological, paratopological and
topological vector spaces.
Definition 2.1. Let L be a vector space that is equipped with a topology τ such that (L, +, τ)
is a paratopological group. We say that (L, τ ) is
1. paratopological vector space if for each neighborhood U of rx with x ∈ L and r ∈ R
+
(the
set of non-negative real numbers), there exist a neighborhood V of x and an > 0 such
that [r, r + [.V ⊆ U;
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
Super quasi-topological and paratopological vector spaces versus topological vector spaces 3
2. quasi-topological vector space if the function ϕ
r
: L → L defined by ϕ
r
(x) = rx with r ∈ R
+
,
is continuous;
3. super quasi-topological vector space if the function ϕ
r
: L → L defined by ϕ
r
(x) = rx with
r ∈ R, is continuous.
Proposition 2.1. There is a first countable locally connected quasi-topological vector space which
is not a super quasi-topological vector space.
Proof. Suppose that the complex vector space C × C is endowed with the topology which has a
base of the sets of the form D
r
×D
s
where D
r
= {
1
√
2
(x−y)+
i
√
2
(x+y): x, y ∈ R, x ≥ r, i
2
= −1},
D
s
= {s + iy : y ∈ R, i
2
= −1} and r, s ∈ R. Then C × C is a first countable locally connected
quasi-topological vector space but it is not a super quasi-topological vector space. Furthermore,
C × C is not a paratopological vector space. Also, it is neither a second countable nor a lindelof
space.
Proposition 2.2. There is a first countable non-connected quasi-topological vector space which
is not a paratopological vector space.
Proof. Endow the complex vector space C with the topology generated by the family of sets of
the form D
r
= {
1
√
2
(x − r) +
i
√
2
(x + r): x ∈ R, i
2
= −1}, with r ∈ R. Then C is first countable
non-connected quasi-topological vector space. Observe that C is not a paratopological vector
space.
Proposition 2.3. There is a first countable connected paratopological vector space which is not
a topological vector space.
Proof. Consider the topology on the complex vector space C × C which has a base of the sets
of the form P
r
× Q
s
, where P
r
= {
1
√
2
(x − y) +
i
√
2
(x + y): x, y ∈ R, x > r, i
2
= −1},
Q
s
= {x + iy : x, y ∈ R, y > s, i
2
= −1} and r, s ∈ R. Then C × C with this topology is a
first countable connected paratopological vector space which is not a topological vector space.
Moreover, it is second countable as well as lindelof space.
Proposition 2.4. There is a first countable non-connected super quasi-topological vector space
which is not a paratopological vector space.
Proof. Obtain the topology on the complex vector space C by the family of sets of the form
Q
r
= {
1
2
(r −
√
3y) +
i
2
(
√
3r + y): y ∈ R, i
2
= −1}, with r ∈ R. Then C with this topology
is a first countable super quasi-topological vector space, but it is not a paratopological vector
space.
Proposition 2.5. There is a first countable connected real quasi-topological vector space which
is not a super quasi-topological vector space.
Proof. Consider the topology on the real vector space R generated by the family of sets of the form
[a, + ∞), with a ∈ R. Then R with this topology is a first countable connected quasi-topological
vector space which is not a super quasi-topological vector space.
Proposition 2.6. Let (L, τ ) be a complex paratopological vector space. Then (L, τ
θ
) is also a
paratopological vector space where τ
θ
= {e
iθ
U : U ∈ τ, 0 ≤ θ ≤ 2π}.
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
4 Madhu Ram - Bijan Davvaz
Proof. Let x and y be any two elements of L, and e
iθ
D an open neighborhood of x + y (with
respect to the topology τ
θ
). Then there exist a neighborhood U of e
−iθ
x and a neighborhood V
of e
−iθ
y (with respect to the topology τ ) such that U + V ⊆ D. As e
−iθ
x ∈ U and e
−iθ
y ∈ V ,
we have x ∈ e
iθ
U and y ∈ e
iθ
V . This gives
x + y ∈ e
iθ
(U + V ) ⊆ e
iθ
D.
Let r be any non-negative real number and e
iθ
U an open neighborhood of rx (with respect to
the topology τ
θ
). Then there exist a neighborhood V of e
−iθ
x (with respect to the topology τ)
and an > 0 such that [r, r + [.V ⊆ U which implies that rx ∈ [r, r + [.e
iθ
V ⊆ [r, r + [.e
iθ
U.
Thus (L, τ
θ
) is a paratopological vector space.
Proposition 2.7. Let (L, τ ) be a complex quasi-topological vector space. Then (L, τ
θ
) is also
a quasi-topological vector space where τ
θ
= {e
iθ
U : U ∈ τ, 0 ≤ θ ≤ 2π}.
Proof. Follows in a similar way as the proof of Proposition 2.6.
Proposition 2.8. Let (L, τ ) be a complex super quasi-topological vector space. Then (L, τ
θ
) is
also a super quasi-topological vector space where τ
θ
= {e
iθ
U : U ∈ τ, 0 ≤ θ ≤ 2π}.
Proof. Follows in a similar way as the proof of Proposition 2.6.
Definition 2.2. We say that a quasi-topological vector space (L, τ ) is strong if it satisfies the
following conditions:
1. there exists a topology = on L such that (L, =) is a topological vector space with = ⊆ τ ,
and
2. there exists a local base B at the zero vector of the quasi-topological vector space (L, τ )
such that V \{0} is open in (L, =) for every V ∈ B.
Proposition 2.9. There exists a first countable non-connected strong quasi-topological vector
space which is not second countable.
Proof. Consider the real vector space R endowed with the topology τ which has a base of the
sets of the form (a, b) and [c, + ∞), where a, b and c are real numbers. Then (R, τ ) is a first
countable strong quasi-topological vector space. Clearly, it is neither a connected space nor a
second countable space.
Proposition 2.10. There exists a first countable non-connected quasi-topological vector space
which is not strong.
Proof. Consider the complex plane C endowed with the topology τ which has a base of the sets
of the form D(z, r) and D
t
where D(z, r) denotes the open disk with center z and radius r, and
D
t
= {z ∈ C : Re(z) ≥ t, t ∈ R}. Then (C, τ) is a quasi-topological vector space which is not
strong.
Proposition 2.11. There exists a regular super quasi-topological vector space which is not strong.
Proof. Let C and τ be as in Proposition 2.5. Then C is not a strong quasi-topological vector
space.
Proposition 2.12. There exists a Hausdorff strong quasi-topological vector space which is not a
super quasi-topological vector space.
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
Super quasi-topological and paratopological vector spaces versus topological vector spaces 5
Proof. Let R and τ be as in Proposition 2.11. Then R is not a super quasi-topological vector
space.
The following result collects the above information and shows that the class of paratopological
vector spaces and the class of quasi-topological vector spaces are sufficiently wide.
Theorem 2.1. The following statements are valid.
1. The class of quasi-topological vector spaces contains the class of super quasi-topological,
strong quasi-topological, paratopological and topological vector spaces.
2. The class of super quasi-topological vector spaces contains the class of topological vector
spaces.
3. The class of super quasi-topological vector spaces is independent of the class of paratopolog-
ical vector spaces.
3 Basic properties of super topological vector spaces
In this section, we investigate some basic properties of super quasi-topological vector spaces. By
definition, every topological vector space is a super quasi-topological vector space, so our results
on a super quasi-topological vector space can be viewed as either improvements or extensions of
results in topological vector spaces. When we say that a topology τ is a super quasi-topology on
a vector space L, we mean that (L, τ) is a super quasi-topological vector space.
Theorem 3.1. For a super quasi-topology τ on a vector space L, x ∈ L and a non-zero real r,
the following hold:
1. the function T
x
: L → L defined by T
x
(y) = x + y is a homeomorphism;
2. the function H
r
: L → L defined by H
r
(x) = rx is a homeomorphism.
Consequently for any subset P of L, we have Cl(x + P ) = x + Cl(P ); Int(x + P ) = x + Int(P );
Cl(rP ) = rCl(P ); Int(rP ) = rInt(P) and for any open (closed) subset Q of L, x + Q and rQ
are open (closed).
Corollary 3.1. Every super quasi-topological vector space is a homogeneous space.
A subset A of a super quasi-topological vector space L is called semi-balanced if for each
x ∈ A, λx ∈ A whenever −1 ≤ λ ≤ 1. It is semi-absorbent if for each x ∈ L, there is a real r > 0
such that λx ∈ A for each real λ satisfying −r < λ < r. Moreover, A is called bounded if for
every neighborhood U of 0, there is a real t > 0 such that A ⊆ sU for all reals s satisfying |s| ≥ t.
As a consequence of Theorem 3.1, it can be shown in a similar way to that of topological
vector spaces, the following result:
Theorem 3.2. Suppose that (L, τ ) is a super quasi-topological vector space, x ∈ L, 0 6= r ∈ R
and A, B are subsets of L. The following assertions are valid:
1. A is open if and only if x + A and rA are open;
2. A is closed if and only if x + A and rA are closed;
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
6 Madhu Ram - Bijan Davvaz
3. A is compact if and only if x + A and rA are compact;
4. if A is convex, then so are Cl(A) and Int(A);
5. if A is semi-balanced, then so is Cl(A);
6. if A and B are compact, then A + B is compact;
7. if A and B are connected, then A + B is connected;
8. if A and B are bounded, then so are Cl(A) and A ∪ B;
9. any finite subset of L is bounded.
Theorem 3.3. Let τ be a super quasi-topology on a vector space L. There exists a local base B
at the origin satisfying the following conditions:
1. for every U, V ∈ B, there is W ∈ B such that W ⊆ U ∩ V ;
2. for each U ∈ B, there is V ∈ B such that V + V ⊆ U ;
3. for each U ∈ B, there is a symmetric V ∈ B such that V + V ⊆ U;
4. for each U ∈ B and for each x ∈ U, there is V ∈ B such that x + V ⊆ U;
5. for each U ∈ B and r ∈ R, there is V ∈ B such that rV ⊆ U and V r ⊆ U.
Conversely, let L be a vector space and let B be a family of subsets of L satisfying (1)-(5) and
that each member of B contains the origin. Then there is a super quasi-topology on L with B as
a base of neighborhoods of the origin.
Proof. From Definition 2.1, and Theorem 3.1, it is easy to check that conditions (1)-(5) hold.
To prove the converse part, let B be a family of subsets of L satisfying the conditions (1)-(5)
and that each member of B contains 0. Let = = {W ⊆ L: for every x ∈ W, there exists U ∈ B
such that x + U ⊆ W }.
Claim 1. = is a topology on L.
Clearly, L ∈ = and ∅ ∈ =. It is also easy to see that = is closed under unions. To show that
= is closed under finite intersections, let P, Q ∈ = and let x ∈ P ∩ Q. Then there exist U, V ∈ B
such that x + U ⊆ P and x + V ⊆ Q. From condition (1), it follows that there exists O ∈ B such
that O ⊆ U ∩ V . Then x + O ⊆ P ∩Q. Hence P ∩ Q ∈ =, and = is a topology on L.
Claim 2. If W ∈ B and x ∈ L, then x + W ∈ =.
Let y ∈ x + W be an arbitrary element. Then −x + y ∈ W . From condition (4), it follows
that there exists U ∈ B such that −x + y + U ⊆ W . This means that y + U ⊆ x + W . Hence
x + W ∈ =.
Claim 3. The family T
B
= {x + U : x ∈ L, U ∈ B} is a base for the topology = on L.
Obviously, it follows from Claim 2.
Claim 4. The vector addition mapping in L is continuous with respect to the topology =.
Let x, y be arbitrary elements of L and let W be an element of = such that x + y ∈ W . Then
there exists U ∈ B such that x + y + U ⊆ W . For U, there is V ∈ B such that V + V ⊆ U by
condition (2). Then x + V and y + B be two elements of T
B
containing x and y, respectively
such that
(x + V ) + (y + V ) ⊆ x + y + V + V ⊆ x + y + U ⊆ W.
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Super quasi-topological and paratopological vector spaces versus topological vector spaces 7
This ends claim 4.
Claim 5. The function H
r
: L → L defined by H
r
(x) = rx is continuous with r ∈ R.
Let W be an element of = containing rx with x ∈ L. Then there exists U ∈ B such that
rx + U ⊆ W . By condition (5), there is V ∈ B such that rV ⊆ U. Then r(x + V ) = rx + rV ⊆
rx + U ⊆ W . This shows that H
r
is continuous.
Theorem 3.4. Let (L, τ ) be a super quasi-topological vector space. If V is the neighborhood filter
of the origin, then for each x ∈ L, F(x) = {x + V : V ∈ V} is the neighborhood filter of the point
x. Consequently, a topology of a super quasi-topological vector space is completely determined by
the neighborhood filter of the origin.
Theorem 3.5. Let (L, τ ) be a super quasi-topological vector space. If N is the neighborhood
filter of the origin, then for every A ⊆ L, Cl(A) =
T
{A + U : U ∈ N}.
Proof. Suppose that x ∈ U + A for each U ∈ N, and let W be a neighborhood of x. By Theorem
3.4, there is a symmetric V ∈ N such that x + V ⊆ W . By assumption, there is some a ∈ A such
that x ∈ a + V . Since V is symmetric, a ∈ A ∩ (x + V ). Thus, x ∈ Cl(A).
Conversely, if x ∈ Cl(A), then every neighborhood U + x, U ∈ N, contains a point of A, so
for some u ∈ U, x + u ∈ A. Without loss of generality, we assume that U is symmetric. Then
x ∈ A + U . It ends the proof.
Theorem 3.6. Let (L, τ) be a super quasi-topological vector space and N the neighborhood filter
of zero in L.
1. The open symmetric neighborhoods of the origin form a fundamental system of neighbor-
hoods of the origin.
2. The closed symmetric neighborhoods of the origin form a fundamental system of neighbor-
hoods of the origin.
Proof. (1) Simple.
(2) If V is a neighborhood of zero, then there is U ∈ N such that U +U ⊆ V . By Theorem 3.6,
Cl(U) ⊆ U + U. Thus, V contains a closed neighborhood of zero. If P is a closed neighborhood
of zero, P ∩(−P ) is a closed symmetric neighborhood of zero contained in V by Theorem 3.1.
Example 3.1. Consider the real vector space C = {x+iy : x, y ∈ R, i
2
= −1} where the addition
and multiplication operation of C are the usual addition and multiplication of complex numbers.
Endow C with the topology which has a base of the sets of the form D
r
= {r+ix : y ∈ R, i
2
= −1},
with r ∈ R (the set of real numbers). Then C with this topology is a super quasi-topological vector
space which is neither a paratopological vector space nor a topological vector space.
Theorem 3.7. Let (L, τ) be a super quasi-topological vector space. Then the following conditions
are equivalent:
1. {0} is closed;
2. {0} is the intersection of neighborhoods of the origin;
3. L is Hausdorff.
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
8 Madhu Ram - Bijan Davvaz
Proof. By Theorem 3.6, (1) and (2) are equivalent. (3) ⇒ (2) is obvious. Let x, y be two
elements of L such that x 6= y. Then x − y 6= 0. By part (2), there is a neighborhood V of
0 such that x − y /∈ U . By Theorem 3.4, there is a symmetric neighborhood V of 0 such that
V + V ⊆ U. Then it is easy to check that x + V and y + V are disjoint neighborhoods of x and
y, respectively. It ends the proof.
Example 3.2. Consider the vector space C as in Example 3.8. For each z
0
∈ C, with y
0
=
Im(z
0
), denote by L
y
0
= {x + iy
0
: x ∈ R, i
2
= −1}, the horizontal line passing through y
0
, and
B
(z
0
), the open ball with center z
0
and radius . Let
U
y
0
, z
0
,
= L
y
0
∩ B
(z
0
) (3.1)
Obtain the topology on C generated by the family of sets of the form (3.1). Then C is a
Hausdorff super quasi-topological vector space which is not a paratopological vector space.
Example 3.3. Let L be the vector space of all continuous functions on (0, 1). For ϕ ∈ L and
> 0, let U(ϕ, ) = {h ∈ L: |h(x) − ϕ(x)| < , for all x ∈ (0, 1)}. Obtain the topology on
L that these sets U(ϕ, ) generate. Then L with this topology is a super quasi-topological vector
space, but not a topological vector space.
Theorem 3.8. If M is a subspace of a super quasi-topological vector space L, then Cl(M) is
a vector subspace of L over the field of reals. Furthermore, if L is a dense vector subspace of a
super quasi-topological vector space E and if M is a vector subspace of L, then the closure of M
in E is a vector subspace of E over the field of reals.
Proof. Follows from Theorem 3.1.
Theorem 3.9. Let (L, τ) be a super quasi-topological vector space. If C is the connected com-
ponent of the origin and r a non-zero real, then
1. x + C and rC are connected for each x ∈ L;
2. C is a vector subspace of L over the field of reals.
Proof. Straightforward.
A topological space X is totally disconnected if for each x ∈ X, the singleton {x} is connected
component of X. By Theorem 3.6, a super quasi-topological vector space is totally disconnected
if and only if {0} is the connected component of 0.
Theorem 3.10. Let ϕ be a linear map from a super quasi-topological vector space L to a super
quasi-topological vector space E, and let V be the neighborhood filter of the origin in L.
1. ϕ is continuous if and only if it is continuous at 0.
2. ϕ is open if and only if for every V ∈ V, ϕ(V ) is a neighborhood of 0 in E.
Proof. Follows from Theorem 3.1.
Theorem 3.11. If a vector subspace M of a super quasi-topological vector space L has an interior
point, then M is open.
Proof. Let x be an element of M and V a neighborhood of 0 in L such that x + V ⊆ M. Then
for any s ∈ M , we have
s + V = (s −x) + (x + V ) ⊆ M.
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Super quasi-topological and paratopological vector spaces versus topological vector spaces 9
4 Quotients of super quasi-topological vector spaces
A super quasi-topology on vector space L clearly induces a topology on any vector subspace of
L making it a super quasi-topological vector space, and unless the contrary is mentioned, we
shall assume that a vector subspace of a super quasi-topological vector space is furnished with
its induced topology.
Let M be a vector subspace of a super quasi-topological vector space L. Then there is the
canonical map π of L onto L/M , which induces a topology on L/M, called the quotient topology.
Given a vector subspace M of a super quasi-topological vector space L and x ∈ L, denote by
π(x) or ˜x, the coset of M that contains x.
Theorem 4.1. If M is a vector subspace of a super quasi-topological vector space L, then the
quotient map π from L onto L/M is linear, continuous and open.
Proof. The continuity and linearity of π are obvious. Let V be an open subset of L. Since the
map x 7→ a + x from L to L, with a ∈ L is a homeomorphism, π
−1
(π(V )) = V + M, an open
subset of L, so π(V ) is open in L/M.
Theorem 4.2. If M is a vector subspace of a super quasi-topological vector space L, then L/M
is a super quasi-topological vector space.
Proof. Let π(x) and π(y) be two elements of L/M, and let U be an open neighborhood of π(x+y).
Then π
−1
(U) is an open neighborhood of x + y in L, so there exist open neighborhoods V
1
and V
2
of x and y, respectively in L such that V
1
+ V
2
⊆ π
−1
(U). Then π(V
1
) + π(V
2
) ⊆ U. By Theorem
4.1, π(V
1
) and π(V
2
) are open sets in L/M and hence the addition map (π(x), π(y)) 7→ π(x + y)
from L/M × L/M to L/M is continuous.
Let r be any real number. We have to show that the map π(x) 7→ π(rx) from L/M to L/M is
continuous. As L is a super quasi-topological vector space, so for any neighborhood U of π(rx),
there exists an open neighborhood V of x in L such that rV ⊆ π
−1
(U). Then rπ(V ) ⊆ U. It
ends the proof.
Theorem 4.3. If V is the neighborhood filter of 0 in a super quasi-topological vector space L, and
if M is a vector subspace of L, then π(V) is the neighborhood filter of
˜
0 for the quotient topology
of L/M.
Proof. By Theorem 4.1, π(V ) is a neighborhood of
˜
0 in L/M for each V ∈ V. Conversely, if U
is a neighborhood of
˜
0 in L/M, then π
−1
(U) is a neighborhood of 0 in L; so there is V ∈ V such
that V ⊆ π
−1
(U). Thus, π(V ) ⊆ U.
Theorem 4.4. Let M be a vector subspace of a super quasi-topological vector space L.
1. L/M is Hausdorff if and only if M is closed.
2. L/M is discrete if and only if M is open.
Proof. Straightforward.
Theorem 4.5. If M and N are vector subspaces of a super quasi-topological vector space L such
that N ⊆ M , then the quotient topology of M/N is identical with the subspace topology of M/N .
Divulgaciones Matem´aticas Vol. 23-24, No. 1-2 (2022-2023), pp. 1–11
10 Madhu Ram - Bijan Davvaz
Proof. Since M is a vector subspace of L, it is a super quasi-topological vector space with the
topology induced by the topology of L. Let ϕ and π be the canonical mappings from M to M/N
and from L to L/N, respectively. Let U be open for the quotient topology of M/N. Then ϕ
−1
(U)
is open in M, so ϕ
−1
(U) = M ∩ V where V is an open subset of L.
Claim: U = (M/N) ∩ π(V ).
Let η ∈ (M/N) ∩ π(V ). Then η = x + N for some x ∈ M and η = v + N for some v ∈ V .
This implies that v − x ∈ N, so v ∈ x + N ⊆ M + N = M . Therefore, v ∈ M ∩ V = ϕ
−1
(U), so
η = v + N ∈ U . Clearly, U ⊆ (M/N) ∩ π(V ) and the claim follows.
Now let A be open in M/N for the topology on M/N induced by the quotient topology of
L/N. Then A = (M/N) ∩B for some open subset B of L/N. Obviously, ϕ
−1
(A) = M ∩π
−1
(B)
is an open subset of M . This means that A is open for the quotient topology of M/N.
Corollary 4.1. If M and N are vector subspaces of a super quasi-topological vector space L, then
the quotient topology on (M + N)/N is identical with the topology on it induced by the quotient
topology of L/N.
Theorem 4.6. Let f be a linear map from a super quasi-topological vector space L to a super
quasi-topological vector space E, and let M be a vector subspace of L that is contained in the
kernel of f. The linear map g from L/M to E satisfying g ◦ π = f is continuous (open) if and
only if f is continuous (open).
Proof. The necessity part follows from Theorem 4.1. Conversely, assume f is continuous. Let U
be a neighborhood of 0 in E. Then g
−1
(U) = π ◦ f
−1
(U), so g is continuous at 0. By Theorem
3.14, g is continuous.
Theorem 4.7. If M is a vector subspace of a super quasi-topological vector space L, and if M
and L/M are both Hausdorff, then L is Hausdorff.
Proof. Let x be an element of L such that x 6= 0 and let x ∈ U for each U ∈ V, the neighborhood
filter of 0 in L. Since M is Hausdorff, x /∈ M . Then x + M and M are two distinct elements of
L/M. As L/M is Hausdorff, there are disjoint open sets A and B for the quotient topology of
L/M containing x + M and M , respectively. By Theorem 3.14, π
−1
(A) is a neighborhood of x
and π
−1
(B) is a neighborhood of 0 in L. By assumption, x ∈ π
−1
(B), so x ∈ π
−1
(A) ∩ π
−1
(B),
a contradiction. By Theorem 3.9, L is Hausdorff.
Theorem 4.8. If M is the connected component of zero in a super quasi-topological vector space
L, and M a vector subspace, then L/M is totally disconnected.
Proof. Let K be a closed subset of L/M such that π
−1
(K) is disconnected. We will show that K
is disconnected. Let A and B be non-empty subsets of π
−1
(K) such that A ∪ B = π
−1
(K) and
A ∩ B = ∅. As for each x ∈ A, x + M is connected subset of π
−1
(K) and hence A = A + M =
π
−1
(π(A)).
Similarly, B = π
−1
(π(B)).
Since π(A) ∩ π(B) = π(A ∩ B) = ∅ and (L/M )\π(A) = π(L\A) which is open, so π(A) is
closed subset of L/M. Similarly, π(B) is closed in L/M. As
π(A) ∪π(B) = π(A ∪ B) = π(π
−1
(K)) = K,
so K is disconnected. Now,
if C is the connected component of zero in L/M , and if there is a point π(x) of L/M such
that π(x) ∈ C and x /∈ M , then π
−1
(C) would be disconnected, which is a contradiction. It ends
the proof.
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Super quasi-topological and paratopological vector spaces versus topological vector spaces 11
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