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.
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 : 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]
- 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
Retrieved from "//en.wikipedia.org/w/index.php?title=Trivial_topology&oldid=1065481448"
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.