Trivial topology

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 T₁ space. Although it has many other useful properties, these do not make up for this one failing.

Because such a space X is not T₁, 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 subspaces of X also have the trivial topology.
• Arbitrary products 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.

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 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.)

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