The trivial topology is the topology with the least possible number of open sets, 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 T_{1} space. Although it has many other useful properties, these do not make up for this one failing.

Because such a space `X` is not T_{1}, it is also not Hausdorff, regular, completely regular, or normal. Because of these, it is not an order topology, and it is not metrizable.

Other properties of a space `X` with the trivial topology, many of which are quite unusual, include:

- The only closed sets are the empty set and
`X`. - The only possible basis of
`X`is {`X`}. -
`X`is compact and therefore paracompact, Lindelöf, and locally compact. - If a function has
`X`as its range, it is continuous. -
`X`is path-connected and so connected. -
`X`is first countable, second countable, and separable. - All subspacess of
`X`also have the trivial topology. - Arbitrary productss of trivial topology spaces, with either the product topology or box topology, have the trivial topology.
- All sequences in
`X`converge to every point of`X`. In particular, every sequence has a convergent subsequence (the whole sequence). - The interior of every set except
`X`is empty. - The closure of every non-empty subset of
*X*is*X*. Put another way: every non-empty subset of*X*is dense, a property that characterizes trivial topological spaces. - If
*S*is any subset of*X*with more than one element, then all elements of*X*are limit points of*S*. If*S*is a singleton, then every point of*X*\\*S*is still a limit point of*S*. -
`X`is a Baire space. - Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.

The trivial topology belongs to a pseudometric space in which the distance between any two points is zero, and 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 *F* : **Top** → **Set** is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and *G* : **Set** → **Top** is the functor that puts the trivial topology on a given set, then *G* is right adjoint to *F*. (The functor *H* : **Set** → **Top** that puts the discrete topology on a given set is *left adjoint* to *F*.)