Basis for indiscrete topology

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, concrete 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 left adjoint of the discrete space functor

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:

The only closed sets are the empty set and X.
The only possible basis of X is {X}.
If X has more than one point, then since it is not T0, it does not satisfy any of the higher 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, completely regular, normal, and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X.
X is compact and therefore paracompact, Lindelöf, and locally compact.
Every function whose domain is a topological space and codomain X is continuous.
X is path-connected and so connected.
X is second-countable, and therefore is first-countable, separable and Lindelöf.
All subspaces of X have the trivial topology.
All quotient spaces of X have the trivial topology
Arbitrary products of trivial topological 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 or any other subsequence), thus X is sequentially compact.
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.
- 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.
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 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 : 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 right adjoint to G. (The so-called free functor F : Set → Top that puts the discrete topology on a given set is left adjoint to G.)[1][2]

List of topologies
Triviality (mathematics)

^ Keegan Smith, "Adjoint Functors in Algebra, Topology and Mathematical Logic", August 8, 2008, p. 13.
^ free functor in nLab

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

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The forgetful functor Γ:Top→Set\Gamma : Top \to Set from Top to Set that sends any topological space to its underlying set has a left adjoint Disc:Set→TopDisc : Set \to Top and a right adjoint Codisc:Set→TopCodisc : Set \to Top.

(Disc⊣Γ⊣Codisc):Top←Codisc→Γ←DiscSet. (Disc \dashv \Gamma \dashv Codisc) : Top \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \,.

For S∈SetS \in Set

Disc(S)Disc(S) is the topological space on SS in which every subset is an open set,

this is called the discrete topology on SS, it is the finest topology on SS; Disc(S)Disc(S) is called a discrete space;
Codisc(S)Codisc(S) is the topological space on SS whose only open sets are the empty set and SS itself, which is called the indiscrete topology on SS (rarely also antidiscrete topology or codiscrete topology or trivial topology or chaotic topology (SGA4-1, 1.1.4)), it is the coarsest topology on SS; Codisc(S)Codisc(S) is called a indiscrete space (rarely also antidiscrete space, even more rarely codiscrete space).

For an axiomatization of this situation see codiscrete object.

Properties

The left adjoint of the discrete space functor

The functor DiscDisc does not preserve infinite products because the infinite product topological space of discrete spaces may be nondiscrete. Thus, DiscDisc does not have a left adjoint functor.

However, if we restrict the codomain of DiscDisc to locally connected spaces, then the left adjoint functor of DiscDisc does exist and it computes the set of connected components of a given locally connected space, i.e., is the π 0\pi_0 functor.

This is discussed at locally connected spaces – cohesion over sets and cosheaf of connected components.

References

For Grothendieck topologies, the terminology “chaotic” is due to

reviewed, e.g., in:

The Stacks Project, Example 7.6.6

Conceptualization of the terminology via right adjoints to forgetful functors (see also at chaos) is due to

William Lawvere, Functorial remarks on the general concept of chaos IMA preprint #87, 1984 (pdf)

and via footnote 1 (page 3) in:

William Lawvere, Categories of spaces may not be generalized spaces, as exemplified by directed graphs, preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 (tac:tr9, pdf).

If $\mathcal{B}$ is a basis, then for each open set $U$ there is a subset $\mathcal{A}\subseteq\mathcal{B}$ such that $U$ is the union of all of the elements of $\mathcal{A}$. In this example, for $U=\emptyset$, you can just take $\mathcal{A}=\emptyset$. The union of all the elements of $\mathcal{A}$ is the union of no sets, which is just the empty set.

By the way, you should say a basis, not the basis, since it's not the only basis for this topology: $\{\emptyset,X\}$ is also a basis.