We study natural topologies in the sense of Debreu in the poset of topological group topologies on a topological group. We call this kind of topologies g-topologies. To be precise, groups admitting a non-totally disconnected g-natural topology as well as the non-totally disconnected g-topologies are identified. Moreover, the non-totally disconnected g-representable topologies as well as the total orders inducing non-totally disconnected group topologies are characterized. It is worth noting that our framework is more general than the usual one in representation theory: we assume no translation invariant properties. We also deal with some questions concerning order and topological algebra related to the semicontinuous representation property (SRP): we establish some results related to the Sorgenfrey line and SRP (some of them connected to the Proper Forcing Axiom (PFA)) and, we characterize σ-compact and (locally) precompact groups which satisfy SRP.
Chis, C., Künzi, H. P. A., & Sanchis, M. (2020). A note on representable group topologies. In Studies in Systems, Decision and Control (Vol. 263, pp. 171–186). Springer. https://doi.org/10.1007/978-3-030-34226-5_8