Editing Trivial topology (section)

==Details==
The trivial topology is the topology with the least possible number of [[open set]]s, 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|T<sub>0</sub> space]].

Other properties of an indiscrete space ''X''&mdash;many of which are quite unusual&mdash;include:

* The only [[closed set]]s are the empty set and ''X''.
* The only possible [[basis (topology)|basis]] of ''X'' is {''X''}.
* If ''X'' has more than one point, then since it is not [[T0 space|T<sub>0</sub>]], it does not satisfy any of the higher [[separation axiom|T axioms]] either. In particular, it is not a [[Hausdorff space]]. Not being Hausdorff, ''X'' is not an [[order topology]], nor is it [[metrizable]].
* ''X'' is, however, [[regular space|regular]], [[completely regular]], [[normal space|normal]], and [[completely normal space|completely normal]]; all in a rather vacuous way though, since the only closed sets are ∅ and ''X''.
* ''X'' is [[compact space|compact]] and therefore [[paracompact]], [[Lindelöf space|Lindelöf]], and [[locally compact]].
* Every [[function (mathematics)|function]] whose [[domain of a function|domain]] is a topological space and [[codomain]] ''X'' is [[continuous function (topology)|continuous]].
* ''X'' is [[path-connected]] and so [[connected space|connected]].
* ''X'' is [[second-countable space|second-countable]], and therefore is [[first-countable space|first-countable]], [[separable space|separable]] and [[Lindelöf space|Lindelöf]].
* All [[subspace (topology)|subspace]]s of ''X'' have the trivial topology.
* All [[Quotient space (topology)|quotient space]]s of ''X'' have the trivial topology
* Arbitrary [[product space|product]]s of trivial topological spaces, with either the [[product topology]] or [[box topology]], have the trivial topology.
* All [[sequence]]s in ''X'' [[limit (mathematics)|converge]] to every point of ''X''. In particular, every sequence has a convergent subsequence (the whole sequence or any other subsequence), thus ''X'' is [[sequentially compact]].
* The [[interior (topology)|interior]] of every set except ''X'' is empty.
* The [[closure (topology)|closure]] of every non-empty subset of ''X'' is ''X''. Put another way: every non-empty subset of ''X'' is [[dense set|dense]], a property that characterizes trivial topological spaces.
** As a result of this, the closure of every open subset ''U'' of ''X'' is either ∅ (if ''U'' = ∅) or ''X'' (otherwise).  In particular, the closure of every open subset of ''X'' is again an open set, and therefore ''X'' is [[Extremally disconnected space|extremally disconnected]].
* If ''S'' is any subset of ''X'' with more than one element, then all elements of ''X'' are [[limit point]]s of ''S''. If ''S'' is a [[singleton (mathematics)|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 [[uniform space]] in which the whole cartesian product ''X'' × ''X'' is the only [[entourage (topology)|entourage]].

Let '''Top''' be the [[category of topological spaces]] with continuous maps and '''Set''' be the [[category of sets]] with functions. 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 [[adjoint functors|right adjoint]] to ''G''. (The so-called [[free functor]] ''F'' : '''Set''' → '''Top''' that puts the [[discrete topology]] on a given set is [[adjoint functors|left adjoint]] to ''G''.)<ref>Keegan Smith, [https://people.cs.uct.ac.za/~ksmith/adjoint.pdf "Adjoint Functors in Algebra, Topology and Mathematical Logic"], August 8, 2008, p. 13.</ref><ref>[https://ncatlab.org/nlab/show/free+functor free functor in nLab]</ref>