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On the topology of free paratopological groups. II

Journal Article


Abstract


  • Let FP(X) be the free paratopological group on a topological space X. For n∈N, denote by FP n(X) the subset of FP(X) consisting of all words of reduced length at most n, and by i n the natural mapping from (X⊕X-1⊕{e})n to FP n(X). In this paper a neighbourhood base at the identity e in FP 2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i2:(X⊕Xd-1⊕{e})2→FP2(X) is a quotient mapping, where X is a T 1 space and Xd-1 denotes the set X -1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T 1 space. Several specific spaces and classes of spaces are also examined. For example, i 2 is a quotient mapping for every countable subspace of R, i 2 is not a quotient mapping for any uncountable compact subspace of R, and it is undecidable in ZFC whether an uncountable subspace of R exists for which i 2 is a quotient mapping. © 2012 Elsevier B.V.

Publication Date


  • 2013

Citation


  • Elfard, A. Sayed. & Nickolas, P. (2013). On the topology of free paratopological groups. II. Topology and its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology, 160 (1), 220-229.

Scopus Eid


  • 2-s2.0-84869222813

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1202

Has Global Citation Frequency


Number Of Pages


  • 9

Start Page


  • 220

End Page


  • 229

Volume


  • 160

Issue


  • 1

Place Of Publication


  • Netherlands

Abstract


  • Let FP(X) be the free paratopological group on a topological space X. For n∈N, denote by FP n(X) the subset of FP(X) consisting of all words of reduced length at most n, and by i n the natural mapping from (X⊕X-1⊕{e})n to FP n(X). In this paper a neighbourhood base at the identity e in FP 2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i2:(X⊕Xd-1⊕{e})2→FP2(X) is a quotient mapping, where X is a T 1 space and Xd-1 denotes the set X -1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T 1 space. Several specific spaces and classes of spaces are also examined. For example, i 2 is a quotient mapping for every countable subspace of R, i 2 is not a quotient mapping for any uncountable compact subspace of R, and it is undecidable in ZFC whether an uncountable subspace of R exists for which i 2 is a quotient mapping. © 2012 Elsevier B.V.

Publication Date


  • 2013

Citation


  • Elfard, A. Sayed. & Nickolas, P. (2013). On the topology of free paratopological groups. II. Topology and its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology, 160 (1), 220-229.

Scopus Eid


  • 2-s2.0-84869222813

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/1202

Has Global Citation Frequency


Number Of Pages


  • 9

Start Page


  • 220

End Page


  • 229

Volume


  • 160

Issue


  • 1

Place Of Publication


  • Netherlands