What is Countability: Definition and 33 Discussions

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.
Some authors use countable set to mean countably infinite alone. To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise.
Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

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  1. BWV

    I Defining a transcendental number and countability

    Is there a term for transcendental numbers that cannot be specified by an operation with a finite amount of data? for example pi or e have various finite definitions and one could generate other transcendental numbers with operations on these. On the other hand if n= some randomly chosen...
  2. fresh_42

    Explore Alan Turing's Computable Numbers & Generate Pi with Python

    I found this article about Alan Turing and his concept of Turing machines on the AMS website. Since we often get questions about countability and computability I thought it is worth sharing. https://blogs.ams.org/featurecolumn/2021/12/01/alan-turing-computable-numbers/ It also contains a Python...
  3. fresh_42

    Challenge Math Challenge - August 2021

    Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus1. Let ##(X,\rho)## be a metric space, and suppose that there exists a sequence ##(f_i)_i## of real-valued continuous functions on ##X## with the property that a...
  4. Z

    I Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?

    Hello experts, Full disclosure: I am a total layman at math, nothing in my training aside from high school courses and one college calculus class. I'm sure a week doesn't pass without someone posting a question about or challenge to Cantor. I am not here to challenge anything but rather to...
  5. D

    I Proof of Countability of ℚ: Bijection from A to ℕ

    I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...
  6. beep300

    I General topology: Countability and separation axioms

    I need some help understanding the countability and separation axioms in general topology, and how they give rise to first-countable and second-countable spaces, T1 spaces, Hausdorff spaces, etc. I more or less get the formal definition, but I can't quite grasp the intuition behind them. Any...
  7. G

    How can the countability of binary trees be proven?

    Homework Statement We'll define a binary tree as a tree where the degree of every internal node is exactly 3. Show that the set of all binary trees is countable. Homework Equations A set is countable if it is finite or there is a one-to-one correspondence with the natural numbers. The Attempt...
  8. A

    Proving the Countability of Nx{0}

    Homework Statement Proove that Nx{0} is countable. x stand for a product i.e. like the cartesian product NxNHomework Equations N is countable.The Attempt at a Solution This is so obvious since Nx{0} is just (1,0),(2,0) etc. But how do you write a proof formally?
  9. C

    What is the true definition of countable sets?

    I seem to have a couple of contradictory statements of what a countable set is defined to be: In my textbook I have: 'Let E be a set. E is said to be countable if and only if there exists a 1-1 function which takes ##\mathbb{N}## onto E.' This implies to me that that there has to exist a...
  10. S

    Proof of Polynomial Countability

    Homework Statement Let P(n) be the set of all polynomial of degree n with integer coefficients. Prove that P(n) is countable, then show that all polynomials with integer coefficients is a countable set. 2. The attempt at a solution For this problem the book gives me a hint that using...
  11. V

    Can countability affect the application of induction?

    I remember my discrete math course in university, where professor told that we can apply induction only to discrete sets. Yet, neither Wikipedia nor Google say nothing about countability importance for induction. They say that underlying set must be well-ordered. The well-ordering topic says...
  12. K

    Minimal number of buckets to hold X marbles while maintaining countability

    You are the owner of a marble warehouse where you store marbles in buckets. You can fit any number of marbles in one bucket. Your job is to store X marbles in a minimal number of buckets. But, when a customer comes and asks for Y number of marbles, you must be able to hand over some buckets...
  13. M

    A short question about countability

    A set is countable if a 1-1 correspondence can be constructed between that set and the set of all positive integers J. Suppose we have a set S consisting of all positive integers plus a "copy" of the element 1: i.e., S={1,1,2,3,4,5,6...}. I have encountered several proofs of basic topological...
  14. W

    Infinite sequence Xn countability

    Homework Statement {xn} is an infinite sequence and xi ≠ xj if i ≠j. Let A and B denote all finite subsequences of {xn} and all infinite subsequences of {xn}, respectively. (a) Show that A is countable. (b) Show that B ≈ (0,1). Homework Equations The Attempt at a...
  15. H

    Countability subset of the reals proof

    Homework Statement Let (a,b)=XUY, X,Y arbitrary sets where (a,b) is an arbitrary interval. Prove that either X or Y has the same cardinality as that of (a,b). Homework Equations The Attempt at a Solution Really lost.
  16. B

    Understanding Countability of Set L: A Confusing Point

    I don't understand this point: Given the open set E = U_(a in L) I_a. Union of open intervals We're showing this is countable. WTS is that indexed set L is countable. Set g: L---> Q (rationals) because Q is dense then every interval meets Q. a---> q_a this is 1-1. But...
  17. K

    Countability of Functions from {0,1}: Finding a 1-1 Correspondence

    Homework Statement Is the set of all functions from {0,1} countable or uncountable? Provide a 1-1 correspondence with a set of know cardinality. Homework Equations The Attempt at a Solution I say it is countable, but my problem is I don't really know how to provide a 1-1...
  18. M

    Countability of Set of Lines Passing Through Two Integer Points

    Homework Statement I have to prove the countability of the set of all lines on the Euclidean plane passing through at least two points whose coordinates are both integers.Homework Equations Proofs don't have particular equations (at least that's what my book says)The Attempt at a Solution First...
  19. radou

    Topological space satisfying 2nd axiom of countability

    Here's another problem which I'd like to check with you guys. So, let X be a topological space which satisfies the second axiom of countability, i.e. there exist some basis B such that its cardinal number is less or equal to \aleph_{0}. One needs to show that such a space is Lindelöf and...
  20. N

    Is the Range of a Countable Function also Countable?

    If the domain of a function is countable, then is its range also countable? also if A is countable and B is countable is A(cartersian product)B countable?
  21. I

    Countability of Set S: Finite vs Denumerable

    A set S is countable if it is either finite or denumerable. What I don't understand is why S can be finite but not denumerable. Could anyone give an example?
  22. C

    Obvoiously True (Countability)

    I'm trying to show that any uncountable set has a countable subset. First, let me point out that the distinction here between at most countable and countable is applied in this instance. At most countable implies either finite or countable, and countable is obvious. Starting off, let X =...
  23. D

    Countability of Algebraic Numbers as Roots of Polynomials

    Homework Statement A real number x \in R is called algebraic if there exist integers a_{0},a_{1},a_{2}...,a_{n}, not all zero, such that a_{n}x^{n} + a_{n}_{1}x^{n-1} + ... + a_{1}x + a_{0} = 0 Said another way, a real number is algebraic if it is the root of a polynomial with integer...
  24. L

    Real Numbers Vector Space: Countability of Basis

    I know that the set of real numbers over the field of rational numbers is an infinite dimensional vector space. BUT I don't quite understand why the basis of that vector space is not countable. Can someone help me?
  25. Q

    Prove Countability: Algebraic Numbers

    So in general, do I always need to use induction to prove that a set is countable? I'm trying to prove that the set of algebraic numbers is countable, but not sure if I am supposed to do it by induction. I am not sure how else to do it.
  26. R

    How Can We Prove the Countability of Algebraic Numbers of Fixed Degree?

    I posted this in the Homework/Coursework section, but I really don't consider it that at all because I'm working through this text on my own, and I'm a little stuck on this problem. Fix n \in N, and let A_n be the algebraic numbers obtained as roots of polynomials with integer coefficients...
  27. R

    Is A_n Countable for Each Fixed n in Algebraic Number Theory?

    Homework Statement Fix n \in N, and let A_n be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree n. Using the fact that every polynomial has a finite number of roots, show that A_n is countable. (For each m \in N, consider the polynomials a_nx^n...
  28. P

    Proof of a Countability Theorem

    Hey all, Can anyone prove this theorem? Let N (natural numbers) ---> X be an onto function. Then X is countable. I've been staring at it for 3 hours and really can't come up with anything. Any help?
  29. S

    Countability of set of sequences

    I was looking at some practice tests and I came upon this tricky question. I'm not sure I would have got it on an exam! Consider the set, S, of all infinite sequences whose entries are either 1 or 2. However, if the nth term is 2 then the n+1th term is 1. I.e every 2 is followed by a one...
  30. M

    What Is First-Countability in a Metric Space: Missed Point Explained

    I have come up with an example when I trying to learn what first countability means It says(from wikipedia) In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis \mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}. This...
  31. S

    Countability: Subjective or Objective?

    "Are there more real numbers between 0 and 1 or between 0 and 2?" If you ask this question to a present day mathematician, he/she would answer that they have the same amount of numbers. Why? Because for every x in the set of numbers between 0 and 2 (call this set A), there is a corresponding...
  32. J

    Help with Countability Questions for Real Numbers

    Hi people, I need some help with these questions please: 1.Is the set of all x in the real numbers such that (x+pi) is rational, countable? I don't think this is countable, isn't the only possible value for x = -pi, all other irrationals will not make x+pi rational i thought? 2.Is...
  33. E

    Does R-omega satisfy the first countability axiom?

    Does R-omega satisfy the first countability axiom? (in the box topology)