In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 with the trivial topology is one where the only open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

s are the empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

 and the entire space. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete topology. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means; it belongs to a pseudometric space

Pseudometric space

In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...

 in which the distance between any two points is zero

0 (number)

0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

.

The trivial topology is the topology with the least possible number of open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

s, 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 set
  Closed set
  In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
  
  s 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
  Separation axiom
  In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
  
   either. In particular, it is not a Hausdorff space
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
  . Not being Hausdorff, X is not an order topology
  Order topology
  In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
  
  , nor is it metrizable.
• X is, however, regular
  Regular space
  In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
  
  , completely regular, normal
  Normal space
  In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
  
  , and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X.
• X is compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   and therefore paracompact, Lindelöf
  Lindelöf space
  In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
  
  , and locally compact.
• Every function
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
   whose domain
  Domain (mathematics)
  In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
  
   is a topological space and codomain
  Codomain
  In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
  
   X is continuous.
• X is path-connected and so connected
  Connected space
  In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
  
  .
• X is second-countable
  Second-countable space
  In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
  
  , and therefore is first-countable
  First-countable space
  In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
  
  , separable and Lindelöf
  Lindelöf space
  In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
  
  .
• All subspaces of X have the trivial topology.
• All quotient space
  Quotient space
  In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
  
  s of X have the trivial topology
• Arbitrary products of trivial topological spaces, with either the product topology
  Product topology
  In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
  
   or box topology
  Box topology
  In topology, the cartesian product of topological spaces can be given several different topologies. The canonical one is the product topology, because it fits rather nicely with the categorical notion of a product. Another possibility is the box topology. The box topology has a somewhat more...
  
  , have the trivial topology.
• All sequence
  Sequence
  In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
  
  s in X converge
  Limit (mathematics)
  In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
  
   to every point of X. In particular, every sequence has a convergent subsequence (the whole sequence), thus X is sequentially compact.
• The interior
  Interior (topology)
  In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
  
   of every set except X is empty.
• The closure
  Closure (topology)
  In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
  
   of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense
  Dense set
  In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
  
  , a property that characterizes trivial topological spaces.
• If S is any subset of X with more than one element, then all elements of X are limit point
  Limit point
  In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
  
  s 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
  Baire space
  In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
  
  .
• Two topological spaces carrying the trivial topology are homeomorphic iff
  IFF
  IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
  
   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

Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...

 in which the whole cartesian product X × X is the only entourage.

Let Top be the category of topological spaces

Category of topological spaces

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...

 with continuous maps and Set be the category of sets

Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

 with functions. If F : Top → Set is the functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

 that assigns to each topological space its underlying set (the so-called forgetful functor

Forgetful functor

In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...

), and G : Set → Top is the functor that puts the trivial topology on a given set, then G is right adjoint

Adjoint functors

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...

 to F. (The functor H : Set → Top that puts the discrete topology on a given set is left adjoint

Adjoint functors

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...

to F.)