The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space X with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space.
Other properties of an indiscrete space Xmany of which are quite unusualinclude:
In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.
The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.
Let Top be the category of topological spaces with continuous maps and Set be the category of sets with functions. If G: Top Set is the functor that assigns to each topological space its underlying set [the so-called forgetful functor], and H: Set Top is the functor that puts the trivial topology on a given set, then H [the so-called cofree functor] is right adjoint to G. [The so-called free functor F: Set Top that puts the discrete topology on a given set is left adjoint to G.][1][2]