Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete.
Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means.
Other properties of an indiscrete space X—many of which are quite unusual—include: 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.
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]