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What is the name of the Topologies that contains only one proper open set? That is if X<>{ } and T={{ }, O,X} ,where O ⊂ X ..I looking for a terminology that represent this topology in general.For me I call them Non-Trivial minimal Topologies .But this is not scientific terminology? please I f there an answer must be supported by a reference ?

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  • $\begingroup$ I would rather call it non-trivial open set. $\endgroup$ – drhab Nov 17 at 14:01
  • $\begingroup$ @drhab I looking for the "official " mathematical terminology for them .If it exist $\endgroup$ – Taha Topology Nov 17 at 14:11
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An infraspace is a topology such that the only topology strictly coarser than it is the trivial topology. Every infraspace topology has on $E$ the form $\{E,A,\emptyset\}$ where $A \subset E$$A \neq \emptyset$, $A \neq E$.  

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS
Volume 5, Number 2, Spring 1975
THE LATTICE OF TOPOLOGIES

This paper first(?) called them infraspaces, same definition. It predates the above paper by 9 years, which makes sense, as the RMJM paper is a survey of earlier results.

So "infraspaces" seems to be sort of accepted, though not widely known, really, as the lattice of topologies is not a very popular subject.

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