Free quasitopological groups

Abstract

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group Fq(X) on a space X. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the “cross topology.” Using this explicit description of Fq(X), we show that for any T1 space X, Fq(X) is the direct limit of closed subspaces Fq(X)n of words of length at most n. We also prove that the natural map in:∐i=0n(X⊔X−1)⊗i→Fq(X)n is quotient for all n⩾0. Equipped with this convenient characterization of the topology of free quasitopological groups, we show, among other things, that a subspace Y⊆X is closed if and only if the inclusion Y→X induces a closed embedding Fq(Y)→Fq(X) of free quasitopological groups.