In mathematics, a **countable set** is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a **countably infinite** set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number.

- A note on terminology
- Definition
- History
- Introduction
- Formal overview
- Minimal model of set theory is countable
- Total orders
- See also
- Notes
- Citations
- References

Georg Cantor introduced the concept of countable sets, contrasting sets that are countable with those that are uncountable. Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal.^{ [1] } Another common style uses *countable* to mean what is here called countably infinite, and *at most countable* to mean what is here called countable.^{ [2] }^{ [3] } To avoid ambiguity, one may limit oneself to the terms "at most countable" and "countably infinite", although with respect to concision this is the worst of both worlds.^{[ citation needed ]} The reader is advised to check the definition in use when encountering the term "countable" in the literature.

The terms *enumerable*^{ [4] } and **denumerable**^{ [5] }^{ [6] } may also be used, e.g. referring to countable and countably infinite respectively,^{ [7] } but as definitions vary the reader is once again advised to check the definition in use.^{ [8] }

The most concise definition is in terms of cardinality. A set S is *countable* if its cardinality |S| is less than or equal to (aleph-null), the cardinality of the set of natural numbers **N**. A set S is *countably infinite * if |S| = . A set is * uncountable * if it is not countable, i.e. its cardinality is greater than ; the reader is referred to Uncountable set for further discussion.^{ [9] }

Equivalently, a set S is countable iff:^{ [5] }

- there exists an injective function from S to
**N**.^{ [10] }^{ [11] } - S is empty or there exists a surjective function from
**N**to S, or .^{ [11] } - there exists a bijective mapping between S and a subset of
**N**.^{ [12] } - S is either finite or countably infinite.
^{ [13] }

Similarly, a set S is countably infinite iff:

- there is an injective and surjective (and therefore bijective) mapping between S and
**N**. In other words, a set is countably infinite if it has one-to-one correspondence with the natural number set,**N**.^{ [14] } - The elements of S can be arranged in an infinite sequence , where is distinct from for and every element of S is listed.
^{ [15] }^{ [16] }

In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable.^{ [17] } In 1878, he used one-to-one correspondences to define and compare cardinalities.^{ [18] } In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.^{ [19] }

A * set * is a collection of *elements*, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}, called roster form.^{ [20] } This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example, {1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still *possible* to list all the elements, because number of elements in the set is finite.

Some sets are *infinite*; these sets have more than *n* elements where *n* is any integer that can be specified. (No matter how large the specified integer *n* is, such as *n* = 9 × 10^{32}, infinite sets have more than *n* elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...},^{ [lower-alpha 1] } has infinitely many elements, and we cannot use any natural number to give its size. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, *cardinality*, the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

To understand what this means, we first examine what it *does not* mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of even integers, which is the same as the number of odd integers, is also the same as the number of integers overall. This is because we can arrange things such that, for every integer, there is a distinct even integer:

or, more generally, (see picture). What we have done here is arrange the integers and the even integers into a *one-to-one correspondence* (or * bijection *), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

By definition, a set *S* is *countable* if there exists an injective function *f* : *S* → **N** from *S* to the natural numbers **N** = {0, 1, 2, 3, ...}. It simply means that every element in *S* has the correspondence to a different element in **N***.*

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size.

To elaborate this, we need the concept of a bijection. Although a "bijection" may seem a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

*a*↔ 1,*b*↔ 2,*c*↔ 3

Since every element of {*a*, *b*, *c*} is paired with *precisely one* element of {1, 2, 3}, *and* vice versa, this defines a bijection.

We now generalize this situation; we *define* that two sets are of the same size, if and only if there is a bijection between them. For all finite sets, this gives us the usual definition of "the same size".

As for the case of infinite sets, consider the sets *A* = {1, 2, 3, ... }, the set of positive integers, and *B* = {2, 4, 6, ... }, the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore *B* is countably infinite. Recall that to prove this, we need to exhibit a bijection between them. This can be achieved using the assignment *n* ↔ 2*n*, so that

- 1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This is an example of a set of the same size as one of its proper subsets, which is impossible for finite sets.

Likewise, the set of all ordered pairs of natural numbers (the Cartesian product of two sets of natural numbers, **N** × **N**) is countably infinite, as can be seen by following a path like the one in the picture:

The resulting mapping proceeds as follows:

- 0 ↔ (0, 0), 1 ↔ (1, 0), 2 ↔ (0, 1), 3 ↔ (2, 0), 4 ↔ (1, 1), 5 ↔ (0, 2), 6 ↔ (3, 0), ....

This mapping covers all such ordered pairs.

This form of triangular mapping recursively generalizes to *n*-tuples of natural numbers, i.e., (*a _{1}*,

**Theorem:** The Cartesian product of finitely many countable sets is countable.^{ [21] }^{ [lower-alpha 2] }

The set of all integers **Z** and the set of all rational numbers **Q** may intuitively seem much bigger than **N**. But looks can be deceiving. If a pair is treated as the numerator and denominator of a vulgar fraction (a fraction in the form of *a*/*b* where *a* and *b* ≠ 0 are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number is also a fraction *N*/1. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below.

**Theorem:****Z** (the set of all integers) and **Q** (the set of all rational numbers) are countable.^{ [lower-alpha 3] }

In a similar manner, the set of algebraic numbers is countable.^{ [23] }^{ [lower-alpha 4] }

Sometimes more than one mapping is useful: a set A to be shown as countable is one-to-one mapped (injection) to another set B, then A is proved as countable if B is one-to-one mapped to the set of natural numbers. For example, the set of positive rational numbers can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because *p*/*q *maps to (*p*, *q*). Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.

**Theorem:** Any finite union of countable sets is countable.^{ [24] }^{ [25] }^{ [lower-alpha 5] }

With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so.

**Theorem:** (Assuming the axiom of countable choice) The union of countably many countable sets is countable.^{ [lower-alpha 6] }

For example, given countable sets **a**, **b**, **c**, ...

Using a variant of the triangular enumeration we saw above:

*a*_{0}maps to 0*a*_{1}maps to 1*b*_{0}maps to 2*a*_{2}maps to 3*b*_{1}maps to 4*c*_{0}maps to 5*a*_{3}maps to 6*b*_{2}maps to 7*c*_{1}maps to 8*d*_{0}maps to 9*a*_{4}maps to 10- ...

This only works if the sets **a**, **b**, **c**, ... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem.

We need the axiom of countable choice to index *all* the sets **a**, **b**, **c**, ... simultaneously.

**Theorem:** The set of all finite-length sequences of natural numbers is countable.

This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.

**Theorem:** The set of all finite subsets of the natural numbers is countable.

The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

**Theorem:** Let *S* and *T* be sets.

- If the function
*f*:*S*→*T*is injective and*T*is countable then*S*is countable. - If the function
*g*:*S*→*T*is surjective and*S*is countable then*T*is countable.

These follow from the definitions of countable set as injective / surjective functions.^{ [lower-alpha 7] }

** Cantor's theorem ** asserts that if *A* is a set and *P*(*A*) is its power set, i.e. the set of all subsets of *A*, then there is no surjective function from *A* to *P*(*A*). A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have:

**Proposition:** The set *P*(**N**) is not countable; i.e. it is uncountable.

For an elaboration of this result see Cantor's diagonal argument.

The set of real numbers is uncountable,^{ [lower-alpha 8] } and so is the set of all infinite sequences of natural numbers.

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standard model (*see* Constructible universe). The Löwenheim–Skolem theorem can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model *M* contains elements that are:

- subsets of
*M*, hence countable, - but uncountable from the point of view of
*M*,

was seen as paradoxical in the early days of set theory, see Skolem's paradox for more.

The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.

Countable sets can be totally ordered in various ways, for example:

- Well-orders (see also ordinal number):
- The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)
- The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)

- Other (
*not*well orders):- The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)
- The usual order of rational numbers (Cannot be explicitly written as an ordered list!)

In both examples of well orders here, any subset has a *least element*; and in both examples of non-well orders, *some* subsets do not have a *least element*. This is the key definition that determines whether a total order is also a well order.

- ↑ Since there is an obvious bijection between
**N**and**N*** = {1, 2, 3, ...}, it makes no difference whether one considers 0 a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which takes 0 as a natural number. - ↑
**Proof:**Observe that**N**×**N**is countable as a consequence of the definition because the function*f*:**N**×**N**→**N**given by*f*(*m*,*n*) = 2^{m}3^{n}is injective.^{ [22] }It then follows that the Cartesian product of any two countable sets is countable, because if A and B are two countable sets there are surjections*f*:**N**→*A*and*g*:**N**→*B*. So*f*×*g*:**N**×**N**→*A*×*B*

**N**×**N**to the set*A*×*B*and the Corollary implies*A*×*B*is countable. This result generalizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction on the number of sets in the collection. - ↑
**Proof:**The integers**Z**are countable because the function*f*:**Z**→**N**given by*f*(*n*) = 2^{n}if n is non-negative and*f*(*n*) = 3^{− n}if n is negative, is an injective function. The rational numbers**Q**are countable because the function*g*:**Z**×**N**→**Q**given by*g*(*m*,*n*) =*m*/(*n*+ 1) is a surjection from the countable set**Z**×**N**to the rationals**Q**. - ↑
**Proof:**Per definition, every algebraic number (including complex numbers) is a root of a polynomial with integer coefficients. Given an algebraic number , let be a polynomial with integer coefficients such that is the*k*th root of the polynomial, where the roots are sorted by absolute value from small to big, then sorted by argument from small to big. We can define an injection (i. e. one-to-one) function*f*:**A**→**Q**given by , while is the*n*-th prime. - ↑
**Proof:**If*A*is a countable set for each i in_{i}*I*={1,...,n}, then for each n there is a surjective function*g*:_{i}**N**→*A*and hence the function_{i}*G*(*i*,*m*) =*g*(_{i}*m*) is a surjection. Since*I*×**N**is countable, the union is countable. - ↑
**Proof**: As in the finite case, but*I*=**N**and we use the axiom of countable choice to pick for each i in**N**a surjection*g*from the non-empty collection of surjections from_{i}**N**to*A*._{i} - ↑
**Proof**: For (1) observe that if*T*is countable there is an injective function*h*:*T*→**N**. Then if*f*:*S*→*T*is injective the composition*h*o*f*:*S*→**N**is injective, so*S*is countable. For (2) observe that if*S*is countable, either*S*is empty or there is a surjective function*h*:**N**→*S*. Then if*g*:*S*→*T*is surjective, either*S*and*T*are both empty, or the composition*g*o*h*:**N**→*T*is surjective. In either case*T*is countable. - ↑ See Cantor's first uncountability proof, and also Finite intersection property#Applications for a topological proof.

- ↑ Manetti, Marco (19 June 2015).
*Topology*. Springer. p. 26. ISBN 978-3-319-16958-3. - ↑ Rudin 1976 , Chapter 2
- ↑ Tao & 2016 181
- ↑ Kamke 1950 , p. 2
- 1 2 Lang 1993 , §2 of Chapter I
- ↑ Apostol 1969 , p. 23, Chapter 1.14
- ↑ Thierry, Vialar (4 April 2017).
*Handbook of Mathematics*. BoD - Books on Demand. p. 24. ISBN 978-2-9551990-1-5. - ↑ Mukherjee, Subir Kumar (2009).
*First Course in Real Analysis*. Academic Publishers. p. 22. ISBN 978-81-89781-90-3. - ↑ Yaqub, Aladdin M. (24 October 2014).
*An Introduction to Metalogic*. Broadview Press. ISBN 978-1-4604-0244-3. - ↑ Singh, Tej Bahadur (17 May 2019).
*Introduction to Topology*. Springer. p. 422. ISBN 978-981-13-6954-4. - 1 2 Katzourakis, Nikolaos; Varvaruca, Eugen (2 January 2018).
*An Illustrative Introduction to Modern Analysis*. CRC Press. ISBN 978-1-351-76532-9. - ↑ Halmos 1960 , p. 91
- ↑ Weisstein, Eric W. "Countable Set".
*mathworld.wolfram.com*. Retrieved 2020-09-06. - ↑ Kamke 1950 , p. 2
- ↑ Dlab, Vlastimil; Williams, Kenneth S. (9 June 2020).
*Invitation To Algebra: A Resource Compendium For Teachers, Advanced Undergraduate Students And Graduate Students In Mathematics*. World Scientific. p. 8. ISBN 978-981-12-1999-3. - ↑ Tao & 2016 182
- ↑ Stillwell, John C. (2010),
*Roads to Infinity: The Mathematics of Truth and Proof*, CRC Press, p. 10, ISBN 9781439865507,Cantor's discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before 1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish between countable and uncountable infinities could not have been imagined.

- ↑ Cantor 1878, p. 242.
- ↑ Ferreirós 2007, pp. 268, 272–273.
- ↑ "What Are Sets and Roster Form?".
*expii*. 2021-05-09. - ↑ Halmos 1960 , p. 92
- ↑ Avelsgaard 1990, p. 182
- ↑ Kamke 1950 , pp. 3–4
- ↑ Avelsgaard 1990 , p. 180
- ↑ Fletcher & Patty 1988 , p. 187

In mathematics, **cardinal numbers**, or **cardinals** for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The *transfinite* cardinal numbers, often denoted using the Hebrew symbol (aleph) followed by a subscript, describe the sizes of infinite sets.

In mathematics, the **cardinality** of a set is a measure of the "number of elements" of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its **size**, when no confusion with other notions of size is possible.

In mathematics, a **finite set** is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,

In mathematics, the **power set** of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(*S*), 𝒫(*S*), *P*(*S*), , ℘(*S*), or 2^{S}. The notation 2^{S}, meaning the set of all functions from S to a given set of two elements, is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.

In mathematics, a **set** is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if and only if they have precisely the same elements.

In mathematics, an **uncountable set** is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

In set theory, **Cantor's diagonal argument**, also called the **diagonalisation argument**, the **diagonal slash argument**, the **anti-diagonal argument**, the **diagonal method**, and **Cantor's diagonalization proof**, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

In set theory, an **infinite set** is a set that is not a finite set. Infinite sets may be countable or uncountable.

An **enumeration** is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration depend on the discipline of study and the context of a given problem.

**Counting** is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term *enumeration* refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.

In mathematics, particularly in set theory, the **aleph numbers** are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph.

In elementary set theory, **Cantor's theorem** is a fundamental result which states that, for any set , the set of all subsets of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with members has a total of subsets, so that if then , and the theorem holds because for all non-negative integers.

In mathematics, two sets or classes *A* and *B* are **equinumerous** if there exists a one-to-one correspondence between them, that is, if there exists a function from *A* to *B* such that for every element *y* of *B*, there is exactly one element *x* of *A* with *f*(*x*) = *y*. Equinumerous sets are said to have the same cardinality. The study of cardinality is often called **equinumerosity** (*equalness-of-number*). The terms **equipollence** (*equalness-of-strength*) and **equipotence** (*equalness-of-power*) are sometimes used instead.

In mathematics, a set *A* is **Dedekind-infinite** if some proper subset *B* of *A* is equinumerous to *A*. Explicitly, this means that there exists a bijective function from *A* onto some proper subset *B* of *A*. A set is **Dedekind-finite** if it is not Dedekind-infinite. Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.

In set theory, the **cardinality of the continuum** is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by or .

This article contains a discussion of **paradoxes of set theory**. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.

In constructive mathematics, a collection is **subcountable** if there exists a partial surjection from the natural numbers onto it. This may be expressed as

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol **R** or and is sometimes called "the reals".

**Cantor's first set theory article** contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using **Cantor's first uncountability proof**, which differs from the more familiar proof using his diagonal argument. The title of the article, "**On a Property of the Collection of All Real Algebraic Numbers**", refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set being dense in an interval.

In set theory, an **ordinal number**, or **ordinal**, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

- Apostol, Tom M. (June 1969),
*Multi-Variable Calculus and Linear Algebra with Applications*, Calculus,**2**(2nd ed.), New York: John Wiley + Sons, ISBN 978-0-471-00007-5 - Avelsgaard, Carol (1990),
*Foundations for Advanced Mathematics*, Scott, Foresman and Company, ISBN 0-673-38152-8 - Cantor, Georg (1878), "Ein Beitrag zur Mannigfaltigkeitslehre",
*Journal für die Reine und Angewandte Mathematik*,**1878**(84): 242–248, doi:10.1515/crelle-1878-18788413 - Ferreirós, José (2007),
*Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought*(2nd revised ed.), Birkhäuser, ISBN 978-3-7643-8349-7 - Fletcher, Peter; Patty, C. Wayne (1988),
*Foundations of Higher Mathematics*, Boston: PWS-KENT Publishing Company, ISBN 0-87150-164-3 - Halmos, Paul R. (1960),
*Naive Set Theory*, D. Van Nostrand Company, Inc Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition). - Kamke, Erich (1950),
*Theory of Sets*, Dover series in mathematics and physics, New York: Dover, ISBN 978-0486601410 - Lang, Serge (1993),
*Real and Functional Analysis*, Berlin, New York: Springer-Verlag, ISBN 0-387-94001-4 - Rudin, Walter (1976),
*Principles of Mathematical Analysis*, New York: McGraw-Hill, ISBN 0-07-054235-X - Tao, Terence (2016). "Infinite sets".
*Analysis I*(Third ed.). Singapore: Springer. pp. 181–210. ISBN 978-981-10-1789-6.

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