Trivial topology

From Academic Kids

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means.

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 T0 space. Although it has many other useful properties, these do not make up for this one failing.

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 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 : TopSet is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and G : SetTop is the functor that puts the trivial topology on a given set, then G is right adjoint to F. (The functor H : SetTop that puts the discrete topology on a given set is left adjoint to F.)

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