2012/06/26 by Ali Sayed Elfard, Elfard, Ali Sayed, Peter Nickolas +1
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras #math.GN
paper · pdf · doi:10.48550/arxiv.1206.5949
This paper has been published by Topology and its Applications
arxiv created 2013/01/12 · arxiv updated 2013/01/15
Let \FP(X) be the free paratopological group on a topological space X. For n∈ \N, denote by \FPn(X) the subset of \FP(X) consisting of all words of reduced length at most n, and by in the natural mapping from (X⊕ X-1⊕ \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 i2\colon (X⊕ X-1d⊕ \e\)2→ \FP2(X) is a quotient map, where X is a T1 space and X-1d denotes the set X-1 equipped with the discrete topology. Further characterisations are given in the case where X is a transitive T1 space. Several specific spaces and classes of spaces are also examined. For example, i2 is a quotient for every countable subspace of \R, i2 is not a quotient for any uncountable compact subspace of \R, and it is undecidable in ZFC whether an uncountable subspace of \R exists for which i2 is a quotient.