The only are the empty set and X.

closed sets

The only possible of X is {X}.

basis

If X has more than one point, then since it is not , 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.

T₀

X is, however, , completely regular, normal, and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X.

regular

X is and therefore paracompact, Lindelöf, and locally compact.

compact

Every whose domain is a topological space and codomain X is continuous.

function

X is and so connected.

path-connected

X is , and therefore is first-countable, separable and Lindelöf.

second-countable

All of X have the trivial topology.

subspaces

All of X have the trivial topology

quotient spaces

Arbitrary of trivial topological spaces, with either the product topology or box topology, have the trivial topology.

products

All 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.

sequences

The of every set except X is empty.

interior

extremally disconnected

If S is any subset of X with more than one element, then all elements of X are of S. If S is a singleton, then every point of X \ S is still a limit point of S.

limit points

X is a .

Baire space

Two topological spaces carrying the trivial topology are iff they have the same cardinality.

homeomorphic

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]

List of topologies

Triviality (mathematics)

; 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