A NOTE ON -SPACES

2014 ◽  
Vol 90 (1) ◽  
pp. 144-148
Author(s):  
HANFENG WANG ◽  
WEI HE
Keyword(s):  
Topological Group ◽  
Positive Answer ◽  
Semitopological Group ◽  
Countable Type ◽  
Countable Tightness

AbstractIn this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.

Property (G), Regularity, and Semi-Equicontinuity

1973 ◽  
Vol 16 (4) ◽  
pp. 587-594
Author(s):  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Function Spaces ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Semitopological Group ◽  
Function Of Two Variables ◽  
The Way

This note, motivated by [2], [3], and [4], is devoted to an investigation of properties related to equicontinuity in function spaces of topological spaces. In §2, we study the property (G) defined in [3], and the regularity defined in [4]. A sufficient condition for the simultaneous continuity of a function of two variables, which is analogous to a well known result in equicontinuity, is given at the end of the section. In §3, we relate the regularity with the semi-equicontinuity defined in [2], by localizing the semi-equicontinuity in an obvious way which leads us to weaken some of the hypotheses used in [2]. By the way of constructing an example, we also obtained a sufficient condition for a regular semitopological group to be a topological group.

THE BAIRE PROPERTY IN THE REMAINDERS OF SEMITOPOLOGICAL GROUPS

2013 ◽  
Vol 88 (2) ◽  
pp. 301-308 ◽  
Author(s):  
LI-HONG XIE ◽  
SHOU LIN
Keyword(s):  
Positive Answer ◽  
Baire Space ◽  
Baire Property ◽  
Topological Groups ◽  
Dichotomy Theorem ◽  
Semitopological Group

AbstractIt is proved that every remainder of a nonlocally compact semitopological group $G$ is a Baire space if and only if $G$ is not Čech-complete, which improves a dichotomy theorem of topological groups by Arhangel’skiǐ [‘The Baire property in remainders of topological groups and other results’, Comment. Math. Univ. Carolin. 50(2) (2009), 273–279], and also gives a positive answer to a question of Lin and Lin [‘About remainders in compactifications of paratopological groups’, ArXiv: 1106.3836v1 [Math. GN] 20 June 2011]. We also show that for a nonlocally compact rectifiable space $G$ every remainder of $G$ is either Baire, or meagre and Lindelöf.

scholarly journals Haar measure and compact right topological groups

1992 ◽  
Vol 45 (3) ◽  
pp. 399-413 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Structure Theorem ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Positive Answer ◽  
Topological Groups ◽  
Low Dimensional ◽  
Compact Right Topological Group

The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.

A note on the continuity of the inverse in paratopological groups

2014 ◽  
Vol 51 (3) ◽  
pp. 326-334 ◽  
Author(s):  
Li-Hong Xie ◽  
Shou Lin
Keyword(s):  
Topological Group ◽  
Locally Compact ◽  
Semitopological Group ◽  
Group 4 ◽  
Countably Compact ◽  
Paratopological Group ◽  
Group 3

The problem when a paratopolgical group (or semitopological group) is a topological group is interesting and important. In this paper, we continue to study this problem. It mainly shows that: (1) Let G be a paratopological group and put τ = ωHs(G); then G is a topological group if G is a Pτ-space; (2) every co-locally countably compact paratopological group G with ωHs(G) ≦ ω is a topological group; (3) every co-locally compact paratopological group is a topological group; (4) each 2-pseudocompact paratopological group G with ωHs(G) ≦ ω is a topological group. These results improve some results in [11, 13].

scholarly journals Positive answers to Koch’s problem in special cases

2020 ◽  
Vol 8 (1) ◽  
pp. 76-87
Author(s):  
Taras Banakh ◽  
Serhii Bardyla ◽  
Igor Guran ◽  
Oleg Gutik ◽  
Alex Ravsky
Keyword(s):  
Topological Group ◽  
Topological Semigroup ◽  
Negative Answer ◽  
Positive Answer ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Special Cases ◽  
Quasitopological Group ◽  
Topological Monoid ◽  
Special Classes

AbstractA topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.

scholarly journals A note on compact-like semitopological groups

2019 ◽  
Vol 11 (2) ◽  
pp. 442-452
Author(s):  
A. Ravsky
Keyword(s):  
Topological Group ◽  
The Other ◽  
Automatic Continuity ◽  
Hausdorff Topology ◽  
Continuity Of Operations ◽  
Semitopological Group ◽  
Separation Axioms ◽  
Other Hand ◽  
Paratopological Group

We present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular $T_1$ compact and $T_2$ sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group $(G,\tau)$ is a topological group provided there exists a Hausdorff topology $\sigma\supset\tau$ on $G$ such that $(G,\sigma)$ is a precompact topological group and $(G,\tau)$ is weakly semiregular or $(G,\sigma)$ is a feebly compact paratopological group and $(G,\tau)$ is $T_3$.

scholarly journals Local compactness in free topological groups

2003 ◽  
Vol 68 (2) ◽  
pp. 243-265 ◽  
Author(s):  
Peter Nickolas ◽  
Mikhail Tkachenko
Keyword(s):  
Topological Group ◽  
Compact Set ◽  
Topological Groups ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Local Compactness ◽  
Countable Type ◽  
Free Abelian Topological Group ◽  
Countable Character

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

scholarly journals NOTES ON -TOPOLOGICAL GROUPS AND HOMEOMORPHISMS OF TOPOLOGICAL GROUPS

2012 ◽  
Vol 87 (3) ◽  
pp. 493-502 ◽  
Author(s):  
HANFENG WANG ◽  
WEI HE
Keyword(s):  
Topological Group ◽  
Affirmative Answer ◽  
Neutral Element ◽  
Topological Groups ◽  
Open Problems ◽  
Urysohn Space ◽  
Compact Topological Group ◽  
Countable Tightness

AbstractIn this paper, it is shown that there exists a connected topological group which is not homeomorphic to any $\omega $-narrow topological group, and also that there exists a zero-dimensional topological group $G$ with neutral element $e$ such that the subspace $X = G\setminus \{e\}$ is not homeomorphic to any topological group. These two results give negative answers to two open problems in Arhangel’skii and Tkachenko [Topological Groups and Related Structures (Atlantis Press, Amsterdam, 2008)]. We show that if a compact topological group is a $K$-space, then it is metrisable. This result gives an affirmative answer to a question posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl. 104 (2000), 181–190] in the category of topological groups. We also prove that a regular $K$-space $X$ is a weakly Fréchet–Urysohn space if and only if $X$has countable tightness.

Sign in / Sign up

Export Citation Format

Share Document