Free paratopological groups

Ali Sayed Elfard

Australia

University of Wollongong

School of Mathematics and Applied Statistics

Submitted: 2013-11-13

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Accepted: 2015-03-08

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Published: 2015-10-01

DOI: https://doi.org/10.4995/agt.2015.1874
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Keywords:

Topological group, paratopological group, free paratopological group, Alexandroff space, partition space, neighborhood base at the identity

Supporting agencies:

This research was not funded

Abstract:

Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a~neighborhood base at the identity of the group FP(X) when the space X is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group FP(X) is T_0 if X is T_0, we characterize the spaces X for which the group FP(X) is a topological group and then we give a class of spaces $X$ for which the group FP(X) has the inductive limit property.
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References:

F. G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenian. (N.S.) 68 (1999), 17-25.

A.V. Arhangel'skii and M. G. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, vol.1, Atlantis Press, Paris, 2008. https://doi.org/10.2991/978-94-91216-35-0_1

A. S. Elfard and P. Nickolas, On the topology of free paratopological groups, Bulletin of the London Mathematical Society 44, no. 6 (2012), 1103-1115. https://doi.org/10.1112/blms/bds031

A. S. Elfard and P. Nickolas, On the topology of free paratopological groups. II, Topology Appl. 160, no. 1 (2013), 220-229. https://doi.org/10.1016/j.topol.2012.10.011

A. S. Elfard, Free paratopological groups, PhD Thesis, University of Wollongong, Australia (2012).

J. Marín and S. Romaguera, A bitopological view of quasi-topological groups, Indian J. Pure Appl. Math. 27 (1996),393-405.

N. M. Pyrch and O.V. Ravsky, On free paratopological groups, Mat. Stud. 25 (2006), 115-125.

N. M. Pyrch, On isomorphisms of the free paratopological groups and free homogeneous spaces I, Visnyk Liviv Univ. Ser. Mech-Math. 63 (2005), 224-232.

N. M. Pyrch, On isomorphisms of the free paratopological groups and free homogeneous spaces II, Visnyk Liviv Univ. 71 (2009), 191-203.

P. Alexandroff, Diskrete Räume, Mat. Sb. (N.S.) 2 (1937), 501-518.

S. Romaguera, M. Sanchis and M. Tkacenko, Free paratopological groups, Proceedings of the 17th Summer Conference on Topology and its Applications 27 (2003), 613-640.

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