A space X is a rotoid if there are a special point e∈X and a homeomorphism H from X2 onto itself such that H(x,x)=(x,e) and H(e,x)=(e,x) for each x∈X. Rotoids are generalizations of topological ...groups, and the Sorgenfrey line is a rotoid and not a topological group. In this paper, we prove some cardinal invariants for rotoids, dichotomy theorems in generalized ordered spaces that are rotoids and dichotomy theorems for remainders in Hausdorff compactifications of paratopological groups that are rotoids. We show, for example, that χ(X)=πχ(X) holds in the class of strong rotoids, that any snf rectifiable space is sof and that any GO-space which is a rotoid is hereditarily paracompact giving an affirmative answer to H. Bennett, D. Burke and D. Lutzerʼs question 11. We also show that any homogeneous GO-space which is a rotoid is first-countable or a totally disconnected P-space, that any remainder of a Hausdorff compactification of a paratopological group which is a rotoid is either pseudocompact or Lindelöf. Moreover, some open questions concerning rotoids are posed.
In this paper, we mainly discuss some generalized metric properties and the character of the free paratopological groups, and extend several results valid for free topological groups to free ...paratopological groups.
Two non-discrete Hausdorff group topologies
τ
and
δ
on a group
G
are called
transversal
if the least upper bound
τ
⋁
δ
of
τ
and
δ
is the discrete topology. In this paper, we discuss the existence of ...transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affirmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko Topology Appl., 2006, 153: 3338–3354. For a compact normal subgroup
K
of a locally compact totally disconnected group
G
, if
G
admits a transversal group topology, then
G/K
admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.
Some results between the properties of strongly topological gyrogroups and the properties of their remainders are established. In particular, if a strongly topological gyrogroup
G
is non-locally ...compact and
G
has a first-countable remainder, then
χ
(
G
)
≤
ω
1
,
ω
(
G
)
≤
2
ω
and
|
b
G
|
≤
2
ω
1
. Moreover, it is proved that the property of paracompact
p
-space of a strongly topological gyrogroup
G
is equivalent with
G
having a Lindelöf remainder in a compactification. By this result, we prove that if
H
is a dense subspace of a strongly topological gyrogroup
G
which is locally pseudocompact and not locally compact, then every remainder of
H
is pseudocompact. Furthermore, if a strongly topological gyrogroup
G
has countable pseudocharacter and
G
is non-metrizable, then all remainders of
G
are pseudocompact. These two results give partial answers to a question posed by Arhangel’ skiǐ and Choban, see (Topol Appl 157:789–799, 2010, Problem 5.1). Finally, it is shown that the Lindelöf property of a non-locally compact strongly topological gyrogroup
G
is equivalent with having a remainder with subcountable type for some compactifications of
G
.
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 investigate copies of Sω and S2 on free topological groups. By applying these results, we show that, for a paracompact space with a point-countable k-network, X is discrete or ...compact if F5(X) is Fréchet–Urysohn, which generalizes Yamada's theorem (Yamada 26). We also give a negative answer to Yamada's conjecture (Yamada 26): If X is a metrizable space, then F4(X) is Fréchet–Urysohn if and only if the set of all non-isolated points of X is compact; and a partial answer to Arhangel'skii's conjecture: Sω1 cannot be embedded into a sequential topological group. Finally, we prove that, for a k⁎-metrizable μ-space X, the free topological group F(X) is a k-space if and only if F5(X) is a k-space. Some questions about free topological groups are posed.
The multiplication of a semitopological (quasitopological) group G is called sequentially continuous if the product map of G×G into G is sequentially continuous. In this paper, we mainly consider the ...properties of semitopological (quasitopological) groups with sequentially continuous multiplications and three-space problems in quasitopological groups. It is showed that (1) every snf-countable semitopological group G with the sequentially continuous multiplication is sof-countable; (2) if G is a sequential quasitopological group with the sequentially continuous multiplication, then G contains a closed copy of Sω if and only if it contains a closed copy of S2, which give a partial answer to a problem posed by R.-X. Shen; (3) let G be a quasitopological group with the sequentially continuous multiplication, then the following are equivalent: (i) G is a sequential α4-space; (ii) G is Fréchet; (iii) G is strongly Fréchet; (4) (MA+¬CH) there exists a non-metrizable, separable, normal and Moore quasitopological group; (5) some examples are constructed to show that metrizability, first-countability and second-countability are not three-space properties in the class of quasitopological groups.
Let (
U
,
R
) be an approximation space with
U
being non-empty set and
R
being an equivalence relation on
U
, and let
G
¯
and
G
̲
be the upper approximation and the lower approximation of subset
G
...of
U
. A topological rough group
G
is a rough group
G
=
(
G
̲
,
G
¯
)
endowed with a topology, which is induced from the upper approximation space
G
¯
, such that the product mapping
f
:
G
×
G
→
G
¯
and the inverse mapping are continuous. In the class of topological rough groups, the relations of some separation axioms are obtained; some basic properties of the neighborhoods of the rough identity element and topological rough subgroups are investigated. In particular, some examples of topological rough groups are provided to clarify some facts about topological rough groups. Moreover, the version of open mapping theorem in the class of topological rough group is obtained. Further, some interesting open questions are posed.
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. In this paper, we mainly discuss the rectifiable spaces which are suborderable, and show that if a rectifiable space is suborderable then it is metrizable or a totally disconnected
P-space, which improves a theorem of A.V. Arhangelʼskiı̌ (2009) in
8. As an application, we discuss the remainders of the Hausdorff compactifications of GO-spaces which are rectifiable, and we mainly concerned with the following statement, and under what condition
Φ it is true.
Statement
Suppose that
G is a non-locally compact GO-space which is rectifiable, and that
Y
=
b
G
∖
G
has (locally) a property-
Φ. Then
G and
bG are separable and metrizable.
Moreover, we also consider some related matters about the remainders of the Hausdorff compactifications of rectifiable spaces.
We mainly discuss the cardinal invariants and generalized metric properties on paratopological groups or rectifiable spaces, and show that: (1) If
A and
B are
ω-narrow subsets of a paratopological ...group
G, then
AB is
ω-narrow in
G, which gives an affirmative answer for A.V. Arhangel'shiı̌ and M. Tkachenko (2008)
7, Open problem 5.1.9; (2) Every bisequential or weakly first-countable rectifiable space is metrizable; (3) The properties of Fréchet–Urysohn and strongly Fréchet–Urysohn coincide in rectifiable spaces; (4) Every rectifiable space
G contains a (closed) copy of
S
ω
if and only if
G has a (closed) copy of
S
2
; (5) If a rectifiable space
G has a
σ-point-discrete
k-network, then
G contains no closed copy of
S
ω
1
; (6) If a rectifiable space
G is pointwise canonically weakly pseudocompact, then
G is a Moscow space. Also, we consider the remainders of paratopological groups or rectifiable spaces, and answer two questions posed by C. Liu (2009) in
20 and C. Liu, S. Lin (2010) in
21, respectively.
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