This is a list of exceptional set concepts. In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and in particular in mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, it is very useful to be able to characterise subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s of a given set X as 'small', in some definite sense, or 'large' if their complement

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

 in X is small. There are numerous concepts that have been introduced to study 'small' or 'exceptional' subsets. In the case of sets of natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

s, it is possible to define more than one concept of 'density', for example. See also list of properties of sets of reals.

• Almost all
  Almost all
  In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
  
  
• Almost always
  Almost Always
  "Almost Always" is a popular song. It was recorded by Joni James in 1953. The recording was released by MGM as catalog number 11470. The song was only on the Billboard magazine charts for one week, reaching #18 on May 9, 1953. The flip side was "Is It Any Wonder?"...
  
  
• Almost everywhere
  Almost everywhere
  In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
  
  
• Almost never
• Almost surely
  Almost surely
  In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
  
  
• Analytic capacity
• Closed unbounded set
  Club set
  In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded relative to the limit ordinal...
  
  
• Cofinal (mathematics)
  Cofinal (mathematics)
  In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition:This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Also, the notion of cofinal...
  
  
• Cofinite
  Cofinite
  In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X...
  
  
• Dense set
  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...
  
  
• IP set
  IP set
  In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D.The set of all finite sums over D is often...
  
  
• 2-large
• Large set (Ramsey theory)
  Large set (Ramsey theory)
  In Ramsey theory, a set S of natural numbers is considered to be a large set if and only if Van der Waerden's theorem can be generalized to assert the existence of arithmetic progressions with common difference in S...
  
  
• Meagre set
  Meagre set
  In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible...
  
  
• Measure zero
  Null set
  In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
  
  
• Natural density
  Natural density
  In number theory, asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is....
  
  
• Negligible set
  Negligible set
  In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.Negligible sets define...
  
  
• Nowhere dense set
  Nowhere dense set
  In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...
  
  
• Null set
  Null set
  In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
  
  , conull set
  Conull set
  In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure.See also:*Almost everywhere...
  
  
• Partition regular
• Piecewise syndetic set
• Schnirelmann density
  Schnirelmann density
  In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician L.G...
  
  
• Small set (combinatorics)
• Stationary set
  Stationary set
  In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:-Classical notion:If \kappa \, is a cardinal of uncountable cofinality, S \subseteq \kappa \,, and S \, intersects every club set in \kappa \,, then S \, is called a stationary set....
  
  
• Syndetic set
  Syndetic set
  In mathematics, a syndetic set is a subset of the natural numbers, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.-Definition:...
  
  
• Thick set
• Thin set (Serre)
  Thin set (Serre)
  In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or...