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**, 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 T_{0} 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 T_{0}, 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.

- As a result of this, the closure of every open subset
- 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] }

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

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In topology and related branches of mathematics, a **topological space** may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis.

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. In a complete metric space, a closed set is a set which is closed under the limit operation.

In topology and related branches of mathematics, a topological space is called **locally compact** if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are *isolated* from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set 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.

In general topology, a branch of mathematics, a collection *A* of subsets of a set *X* is said to have the **finite intersection property** (FIP) if the intersection over any finite subcollection of *A* is non-empty. It has the **strong finite intersection property** (SFIP) if the intersection over any finite subcollection of *A* is infinite.

In mathematics, the **Sierpiński space** is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

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, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

In topology, a **compactly generated space** is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space *X* is compactly generated if it satisfies the following condition:

In topology, a **second-countable space**, also called a **completely separable space**, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the **second axiom of countability**. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In mathematics, a **locally compact group** is a topological group *G* for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on *G* so that standard analysis notions such as the Fourier transform and spaces can be generalized.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" any additional structure the manifold has.

In mathematics, the **particular point topology** is a topology where a set is open if it contains a particular point of the topological space. Formally, let *X* be any set and *p* ∈ *X*. The collection

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of A is constituting the whole set X. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

- 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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