On the topology of free paratopological groups. II

作者：Elfard Ali Sayed^*; Nickolas Peter

来源：Topology and Its Applications, 2013, 160(1): 220-229.

DOI：10.1016/j.topol.2012.10.011

• Paratopological group
• Free paratopological group
• Free topological group
• Quasi-uniform space
• Quasi-uniformity
• Fine quasi-uniformity
• Quasi-pseudometric
• Neighbournet
• Normal neighbournet
• Transitive space
• Graev extension
• Joiner%26apos
• s lemma

Let FP(X) be the free paratopological group on a topological space X. For n is an element of N, denote by FPn(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 circle plus X-1 circle plus {e})(n) to FPn(X). In this paper a neighbourhood base at the identity e in FP2(X) is found. A number of characterisations are then given of the circumstances under which the natural mapping i(2) : (X circle plus X-d(-1) circle plus {e})(2) -%26gt; FP2(X) is a quotient mapping, where X is a T-1 space and X-d(-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.

• 出版日期2013-1-1