Free Quasitopological Groups
Jeremy Brazas, Sarah Emery

TL;DR
This paper investigates the topological structure of free quasitopological groups, providing explicit constructions, limit descriptions, and characterizations of subspace embeddings, advancing understanding of their properties and relationships to the underlying space.
Contribution
It introduces a direct quotient space construction of free quasitopological groups and characterizes their topology via direct limits and quotient maps, offering new insights into their structure.
Findings
Free quasitopological groups can be constructed as quotient spaces of free semitopological monoids.
The topology of $F_q(X)$ is the direct limit of subspaces $F_q(X)_n$ for $T_1$ spaces.
A subspace $Y$ is closed in $X$ iff $F_q(Y)$ embeds as a closed subspace in $F_q(X)$.
Abstract
In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group on a space . 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 , we show that for any space , is the direct limit of closed subspaces of words of length at most . We also prove that the natural map is quotient for all . Equipped with this convenient characterization of the topology of free quasitopological groups, we show, among other things, that a subspace is closed if and only if the inclusion…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Fuzzy and Soft Set Theory · Advanced Algebra and Logic
