Subgroups of paratopological groups and feebly compact groups

Manuel Fernández; Mikhail Tkachenko

doi:10.4995/agt.2014.3157

Authors

Manuel Fernández Universidad Autónoma de la Ciudad de México
Mikhail Tkachenko Universidad Autónoma Metropolitana

DOI:

https://doi.org/10.4995/agt.2014.3157

Keywords:

feebly compact, precompact, paratopological group, subsemigroup, topologically periodic

Abstract

It is shown that if all countable subgroups of a semitopological group G are precompact, then G is also precompact and that the closure of an arbitrary subgroup of G is again a subgroup. We present a general method of refining the topology of a given commutative paratopological group G such that the group G with the finer topology, say, σ is again a paratopological group containing a subgroup whose closure in (G, σ) is not a subgroup.

It is also proved that a feebly compact paratopological group H is perfectly k-normal and that every Gδ-dense subspace of H is feebly compact.

Downloads

Download data is not yet available.

References

A. V. Arhangel'skii and E. A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl. 151 (2005), 107-119.

(http://dx.doi.org/10.1016/j.topol.2003.08.035)

A.V. Arhangel'skii and M. G. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, Vol.1, Atlantis Press and World Scientific, Paris-Amsterdam, 2008.

(http://dx.doi.org/10.2991/978-94-91216-35-0)

R. L. Blair, Spaces in which special sets are $z$-embedded, Canad. J. Math. 28, no. 4 (1976), 673-690.

(http://dx.doi.org/10.4153/CJM-1976-068-9)

T. Banakh and O. Ravsky, On subgroups of saturated or totally bounded paratopological groups, Algebra Discrete Math. 2003, no.4 (2003), 1-20.

T. Banakh and O. Ravsky, Oscillator topologies on a paratopological group and related number invariants, Algebraic Structures and their Applications, Kyiv: Inst. Mat. NANU (2002), 140-152.

M.Fernández, On some classes of paratopological groups, Topology Proc. 40 (2012), 63-72.

O. Ravsky, Paratopological groups, II, Matematychni Studii, 17 (2002) 93-101.

O. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis, Lviv University, 2003 (in Ukrainian).

O. Ravsky, Pseudocompact paratopological groups, arXiv:1003.5343 [Math. GN], September 2013.

E. A. Reznichenko, Extensions of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topolology Appl. 59 (1994), 233-244.

(http://dx.doi.org/10.1016/0166-8641(94)90021-3)

S. Romaguera, M. Sanchis and M. Tkachenko, Free paratopological groups, Topology Proc. 27, no. 2 (2003), 613-640.

M. G. Tkachenko, Paratopological and Semitopological Groups vs Topological Groups, Ch.20 in: Recent Progress in General Topology III (K.P.Hart, J.vanMill, P.Simon, Eds.), Atlantis Press, 2014; pp.825-882.

(http://dx.doi.org/10.2991/978-94-6239-024-9_20)

M. G. Tkachenko, G. Delgadillo Piñón and E. Rodríguez Cervera,

A property of powers of the Sorgenfrey line, Q&A in General Topology 27, no. 1 (2009), 45-49.

M. G. Tkachenko and A. H. Tomita, Cellularity in subgroups of paratopological

groups, preprint.

L.-H. Xie, S. Lin and M. Tkachenko, Factorization properties of paratopological

groups, Topology Appl. 160 (2013), 1902-1917.

(http://dx.doi.org/10.1016/j.topol.2013.08.001)