*"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.*

In mathematical analysis and related areas of mathematics, a set is called **bounded** if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called **unbounded**. The word 'bounded' makes no sense in a general topological space without a corresponding metric.

A set *S* of real numbers is called *bounded from above* if there exists some real number *k* (not necessarily in *S*) such that *k* ≥ * s* for all *s* in *S*. The number *k* is called an **upper bound** of *S*. The terms *bounded from below* and **lower bound** are similarly defined.

A set *S* is **bounded** if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.

A subset *S* of a metric space (*M*, *d*) is **bounded** if there exists *r* > 0 such that for all *s* and *t* in *S*, we have d(*s*, *t*) < *r*. (*M*, *d*) is a *bounded* metric space (or *d* is a *bounded* metric) if *M* is bounded as a subset of itself.

- Total boundedness implies boundedness. For subsets of
**R**^{n}the two are equivalent. - A metric space is compact if and only if it is complete and totally bounded.
- A subset of Euclidean space
**R**^{n}is compact if and only if it is closed and bounded.

In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

A set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size".

A subset *S* of a partially ordered set *P* is called **bounded above** if there is an element *k* in *P* such that *k* ≥ *s* for all *s* in *S*. The element *k* is called an **upper bound** of *S*. The concepts of **bounded below** and **lower bound** are defined similarly. (See also upper and lower bounds.)

A subset *S* of a partially ordered set *P* is called **bounded** if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set *S* but also one of the set *S* as subset of *P*.

A **bounded poset***P* (that is, by itself, not as subset) is one that has a least element and a greatest element. Note that this concept of boundedness has nothing to do with finite size, and that a subset *S* of a bounded poset *P* with as order the restriction of the order on *P* is not necessarily a bounded poset.

A subset *S* of **R**^{n} is bounded with respect to the Euclidean distance if and only if it bounded as subset of **R**^{n} with the product order. However, *S* may be bounded as subset of **R**^{n} with the lexicographical order, but not with respect to the Euclidean distance.

A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as a subclass of the class of all ordinal numbers.

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 **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematics, especially order theory, a **partially ordered set** formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word *partial* in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

In mathematics, a **total order**, **simple order**, **linear order**, **connex order**, or **full order** is a binary relation on some set , which is antisymmetric, transitive, and a connex relation. A set paired with a total order is called a **chain**, a **totally ordered set**, a **simply ordered set**, a **linearly ordered set**, or a **loset**.

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.

In mathematics, a **well-order** on a set *S* is a total order on *S* with the property that every non-empty subset of *S* has a least element in this ordering. The set *S* together with the well-order relation is then called a **well-ordered set**. In some academic articles and textbooks these terms are instead written as **wellorder**, **wellordered**, and **wellordering** or **well order**, **well ordered**, and **well ordering**.

In mathematics, particularly in topology, an **open set** is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets in the collection is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.

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

In mathematics, the **real line**, or **real number line** is the line whose points are the real numbers. That is, the real line is the set **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

In real analysis the **Heine–Borel theorem**, named after Eduard Heine and Émile Borel, states:

In mathematics, a **ball** is the volume space bounded by a sphere; it is also called a **solid sphere**. It may be a **closed ball** or an **open ball**.

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**.

**Domain theory** is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called **domains**. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to topology.

**Order theory** is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.

In the mathematical area of order theory, **completeness properties** assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist.

In mathematics, a **real coordinate space** of dimension n, written **R**^{n} or ℝ^{n}, is a coordinate space over the real numbers. This means that it is the set of the n-tuples of real numbers. With component-wise addition and scalar multiplication, it is a real vector space.

In topology and related branches of mathematics, **total-boundedness** is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size"

In functional analysis and related areas of mathematics, a set in a topological vector space is called **bounded** or **von Neumann bounded**, if every neighborhood of the zero vector can be *inflated* to include the set. A set that is not bounded is called **unbounded**.

In mathematical optimization, **ordinal optimization** is the maximization of functions taking values in a partially ordered set ("poset"). Ordinal optimization has applications in the theory of queuing networks.

This is a **glossary of set theory**.

- Bartle, Robert G.; Sherbert, Donald R. (1982).
*Introduction to Real Analysis*. New York: John Wiley & Sons. ISBN 0-471-05944-7. - Richtmyer, Robert D. (1978).
*Principles of Advanced Mathematical Physics*. New York: Springer. ISBN 0-387-08873-3.

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