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A note on representable group topologies. (English) Zbl 07273625
Bosi, Gianni (ed.) et al., Mathematical topics on representations of ordered structures and utility theory. Essays in honor of Professor Ghanshyam B. Mehta. Cham: Springer. Stud. Syst. Decis. Control 263, 171-186 (2020).
Summary: 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 $$\sigma$$-compact and (locally) precompact groups which satisfy SRP.
For the entire collection see [Zbl 1446.91009].
##### MSC:
 22-XX Topological groups, Lie groups 54-XX General topology
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