For a topological space
X
, let
CL
(
X
) be the set of all non-empty closed subset of
X
, and denote the set
CL
(
X
) with the Vietoris topology by
(
C
L
(
X
)
,
V
)
. In this paper, we mainly ...discuss the hyperspace
(
C
L
(
X
)
,
V
)
when
X
is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of
n
th power of Sorgenfrey line for each
n
∈
N
. Then we investigate the tightness of the hyperspace
(
C
L
(
X
)
,
V
)
and prove that the tightness of
(
C
L
(
X
)
,
V
)
is equal to the set-tightness of
X
. Moreover, we extend some results about the generalized metric properties on the hyperspace
(
C
L
(
X
)
,
V
)
. Finally, we give a characterization of
X
such that
(
C
L
(
X
)
,
V
)
is a
γ
-space.
Suitable sets for paratopological groups Lin, Fucai; Ravsky, Alex; Shi, Tingting
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas,
10/2021, Volume:
115, Issue:
4
Journal Article
Peer reviewed
Open access
A subset
S
of a paratopological group
G
is a
suitable set
for
G
, if
S
is a discrete subspace of
G
,
S
∪
{
e
}
is closed, and the subgroup
⟨
S
⟩
of
G
generated by
S
is dense in
G
. Suitable sets in ...topological groups were studied by many authors. The aim of the present paper is to provide a start-up for a general investigation of suitable sets for paratopological groups, looking to what extent we can (by proving propositions) or cannot (by constructing examples) generalize to paratopological groups results which hold for topological groups, and to pose a few challenging questions for possible future research. We shall discuss when paratopological groups of different classes have suitable sets. Namely, we consider paratopological groups (in particular, countable) satisfying different separation axioms, paratopological groups which are compact-like spaces, and saturated (in particular, precompact) paratopological groups. Also we consider when a property of a group to have a suitable set is preserved with respect to (open or dense) subgroups, products and extensions.
Countable tightness and G-bases on free topological groups Lin, Fucai; Ravsky, Alex; Zhang, Jing
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas,
2020/4, Volume:
114, Issue:
2
Journal Article
Peer reviewed
Open access
Given a Tychonoff space
X
, let
F
(
X
) and
A
(
X
) be respectively the free topological group and the free Abelian topological group over
X
in the sense of Markov. In this paper, we consider two ...topological properties of
F
(
X
) or
A
(
X
), namely the countable tightness and
G
-base. We provide some characterizations of the countable tightness and
G
-base of
F
(
X
) and
A
(
X
) for various special classes of spaces
X
. Furthermore, we also study the countable tightness and
G
-base of some
F
n
(
X
)
of
F
(
X
).
We discuss the class of paratopological groups which admits a transversal, T1-independent and T1-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a ...T1-complementary Hausdorff paratopological group topology, which gives a negative answer to 5, Problem 10. We give a very useful criterion for transversality in term of submaximal paratopological group topology, and prove that if a non-discrete paratopological group topology G contains a central subgroup which admits a transversal paratopological group topology, then so does G. We introduce the concept of PT-sequence and give a characterization of an Abelian paratopological group being determined by a PT-sequence. As the applications, we prove that the Abelian paratopological group, which is endowed with the strongest paratopological group topology being determined by a T-sequence, does not admit a T1-complementary Hausdorff paratopological group topology on G. Finally, we study the class of countable paratopological groups which is determined by a PT-filter, and obtain a sufficient condition for a countable paratopological group G being determined by a PT-sequence which admits a transversal paratopological group topology on G being determined by a PT-sequence.
The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background of c-ball of relativistically admissible velocities with the Einstein ...velocity addition. A topological gyrogroup is just a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. This concept generalizes that of topological groups. In this paper, we are going to establish that for a locally compact admissible L-subgyrogroup H of a strongly topological gyrogroup G, the natural quotient mapping π from G onto the quotient space G/H has some nice local properties, such as, local compactness, local pseudocompactness, and local paracompactness, etc. Finally, we prove that each locally paracompact strongly topological gyrogroup is paracompact.
For a space X, let (CL(X),τV), (CL(X),τlocfin) and (CL(X),τF) be the set CL(X) of all nonempty closed subsets of X which are endowed with Vietoris topology, locally finite topology and Fell topology ...respectively. We prove that (CL(X),τV) is quasi-metrizable if and only if X is a separable metrizable space and the set of all non-isolated points of X is compact, (CL(X),τlocfin) is quasi-metrizable or symmetrizable if and only if X is metrizable and the set of all non-isolated points of X is compact, and (CL(X),τF) is quasi-metrizable if and only if X is hemicompact and metrizable. As an application, we give a negative answer to a Conjecture in 13.
Assume that P is a topological property of a space X, then we say that X is dense-P if each dense subset of X has the property P. In this paper, we mainly discuss dense subsets of a space X, and we ...prove that:
(1) if X is Tychonoff space, then X is dense-pseudocompact if and only if the range of each continuous real-valued function f on X is finite, if and only if X is finite, if and only if X is hereditarily pseudocompact;
(2) X is dense-connected if and only if U‾=X for any non-empty open subset U of X;
(3) X is dense-ultraconnected if and only if for point x∈X, we have {x}‾=X or {x}∪(X∖{x}‾) is the unique open neighborhood of x in {x}∪(X∖{x}‾), if and only if for any two points x and y in X, we have x∈{y}‾ or y∈{x}‾.
Moreover, we give a characterization of a topological group (resp., paratopological group, quasi-topological group) G such that G is dense-connected.
For a regular space X, the hyperspace (CL(X),τF) (resp., (CL(X),τV)) is the space of all nonempty closed subsets of X with the Fell topology (resp., Vietoris topology). In this paper, we give the ...characterization of the space X such that the hyperspace (CL(X),τF) (resp., (CL(X),τV)) with a countable character of closed subsets. We mainly prove that (CL(X),τF) has a countable character on each closed subset if and only if X is compact metrizable, and (CL(X),τF) has a countable character on each compact subset if and only if X is locally compact and separable metrizable. Moreover, we prove that (K(X),τV) have the compact-Gδ-property if and only if X have the compact-Gδ-property and every compact subset of X is metrizable.
PDF
