Document Type

Article

Publication Date

3-2023

Abstract

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group F-q(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 F-q(X), we show that for any T-1 space X, F-q(X) is the direct limit of closed subspaces F-q(X)(n) of words of length at most n. We also prove that the natural map i(n): (sic)(n)(i=0)(X boolean OR X-1)(circle times i) -> F-q(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 subset of X is closed if and only if the inclusion Y -> X induces a closed embedding F-q(Y) -> F-q(X) of free quasitopological groups.

Publication Title

Topology and Its Applications

DOI

10.1016/j.topol.2023.108416

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