Trivial topology

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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 spaces are commonly called indiscrete, anti-discrete, or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

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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 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.

Let Top be the category of topological spaces with continuous maps and Set be the category of sets with functions. If G : TopSet is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and H : SetTop 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 : SetTop that puts the discrete topology on a given set is left adjoint to G.)[1][2]

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  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446