TITLE: Topology with only one open proper set
QUESTION [5 upvotes]: 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 ?

REPLY [9 votes]: 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.