A topological space
G
is said to be a
rectifiable space
provided that there are a surjective homeomorphism
φ
:
G
×
G
→
G
×
G
and an element
e
∈
G
such that
π
1
∘
φ
=
π
1
and for every
x
∈
G
we have
φ
...(
x
,
x
)=(
x
,
e
), where
π
1
:
G
×
G
→
G
is the projection to the first coordinate. Let
G
be a rectifiable space and
C
(
G
) be the family of all non-empty compact subsets of
G
. In this paper, we study the Vietoris topology on
C
(
G
), and show that if
G
is a locally compact rectifiable space, then (
C
(
G
),⋅) together with the Vietoris topology is a topological semi-right loop.
The free topological vector space V(X) over a Tychonoff space X is a pair consisting of a topological vector space V(X) and a continuous mapping i=iX:X→V(X) such that every continuous mapping f from ...X to a topological vector space E gives rise to a unique continuous linear operator f‾:V(X)→E with f=f‾∘i. In this paper, the k-property and countable tightness of free topological vector space over some generalized metric spaces are studied. We mainly discuss the characterizations of a space X such that V(X) or the fourth level of V(X) is a k-space or is of countable tightness, respectively.
The theory of knowledge spaces (KST) which is regarded as a mathematical framework for the assessment of knowledge and advices for further learning. Now the theory of knowledge spaces has many ...applications in education. From the topological point of view, we discuss the language of the theory of knowledge spaces by the axioms of separation and the accumulation points of pre-topology respectively, which establishes some relations between topological spaces and knowledge spaces; in particular, we show that the language of the regularity of pre-topology in knowledge spaces and give a characterization for knowledge spaces by inner fringe of knowledge states. Moreover, we study the relations of Alexandroff spaces and quasi ordinal spaces; then we give an application of the density of pre-topological spaces in primary items for knowledge spaces, which shows that one person in order to master an item, she or he must master some necessary items. In particular, we give a characterization of a skill multimap such that the delineated knowledge structure is a knowledge space, which gives an answer to a problem in 14 or 18 whenever each item with finitely many competencies; further, we give an algorithm to find the set of atom primary items for any finite knowledge space.
In this paper, we continue the study of the symmetric products of generalized metric spaces in 39. We consider the topological properties P such that the n-fold symmetric product Fn(X) of a ...topological space X has the topological properties P if and only if the space X or the product Xn does for each or some n∈N. Depending on the operations under closed subspaces, finite products and closed finite-to-one mappings, two general stability theorems are obtained on symmetric products. We can apply the methods to unify and simplify the proofs of some old results in the literature and obtain some new results on symmetric products, list or prove 68 topological properties which satisfy the general stability theorems, and give answers to Questions 3.6 and 3.35 in 39.
On paratopological groups Lin, Fucai; Liu, Chuan
Topology and its applications,
06/2012, Volume:
159, Issue:
10-11
Journal Article
Peer reviewed
Open access
In this paper, we firstly construct a Hausdorff non-submetrizable paratopological group G in which every point is a Gδ-set, which gives a negative answer to Arhangelʼskiı̌ and Tkachenkoʼs question ...A.V. Arhangelʼskiı̌, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press and World Sci., 2008. We also prove that each first-countable Abelian paratopological group is submetrizable. Moreover, we discuss developable paratopological groups and construct a non-metrizable, separable, Moore paratopological group. Further, we prove that a regular, countable, locally kω-paratopological group is a discrete topological group or contains a closed copy of Sω. Finally, we discuss some properties on non-H-closed paratopological groups, and show that Sorgenfrey line is not H-closed, which gives a negative answer to Arhangelʼskiı̌ and Tkachenkoʼs question A.V. Arhangelʼskiı̌, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press and World Sci., 2008. Some questions are posed.
In this paper, we describe in internal terms the kernel of the canonical homomorphism φG,2 of a semitopological group G onto the T2-reflection T2(G) of G, which answers a problem of M. Tkachenko.
A space X is called a kR-space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is ...continuous. In this paper, we discuss the kR-property of products of sequential fans and free Abelian topological groups by applying the κ-fan introduced by Banakh. In particular, we prove the following two results:(1)The space Sω1×Sω1 is not a kR-space.(2)The space Sω×Sω1 is a kR-space if and only if Sω×Sω1 is a k-space if and only if b>ω1.
These results generalize some well-known results on sequential fans. Furthermore, we generalize some results of Yamada on the free Abelian topological groups by applying the above results. Finally, we pose some open questions about the kR-spaces.
PDF
