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Topic: Countable

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  Countable set - Wikipedia, the free encyclopedia
In mathematics the term countable is used to describe the size of a set, i.e.
The notion of an infinite set is not elementary; it requires a strong sense of abstraction and precision.
A set is called countable if the number of elements is finite or if it has the same number of elements as the natural numbers.
en.wikipedia.org /wiki/Countable   (1069 words)

 PlanetMath: Baire category theorem
In a non-empty complete metric space, any countable intersection of dense, open subsets is non-empty.
In functional analysis, this important property of complete metric spaces forms the basis for the proofs of the important principles of Banach spaces: the open mapping theorem and the closed graph theorem.
It may also be taken as giving a concept of ``small sets'', similar to sets of measure zero: a countable union of these sets remains ``small''.
planetmath.org /encyclopedia/BaireCategoryTheorem.html   (423 words)

 PlanetMath: Banach spaces of infinite dimension don't have a countable algebraic basis
A Banach space of infinite dimension does not have a countable algebraic basis.
Using our result, it follows directly that there is no way to define a norm in this vector space such that it is a complete metric space under the induced metric.
This is version 10 of Banach spaces of infinite dimension don't have a countable algebraic basis, born on 2005-01-31, modified 2005-09-18.
planetmath.org /encyclopedia/ABanachSpaceOfInfiniteDimensionDoesntHaveACountableAlgebraicBasis.html   (448 words)

 PlanetMath: proof that the rationals are countable
"proof that the rationals are countable" is owned by alozano.
Cross-references: union, uncountable, irrational numbers, corollary, countable, bijection, natural number, numerators, classes, rationals, finite, height, lowest terms, rational number
This is version 5 of proof that the rationals are countable, born on 2001-11-16, modified 2002-02-25.
planetmath.org /encyclopedia/ProofThatTheRationalsAreCountable.html   (145 words)

 [No title]
If A is a countable set, and B is a finite set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Answer.
If A is a countable set, and B is an uncountable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Comment.
If A is a countable set, and B is a countable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Answer.
www.math.ucla.edu /~tao/java/MultipleChoice/countable.txt   (804 words)

 The Prime Glossary: infinite   (Site not responding. Last check: 2007-10-22)
Infinite sets are divided into two types: those which are equivalent to a subset of the integers (called countable) and those which are not (called uncountable).
The primes, composites, and positive integers are clearly countable.
But so are the rational numbers, the set of polynomials with integer coefficients, and hence the algebraic numbers.
primes.utm.edu /glossary/page.php?sort=Infinite   (277 words)

 Countable Income - Food Stamps
Countable income is income that is considered in some way when determining gross income, net income, and benefits.
Income that is earned by an individual through receipt of wages, salary, commission, or profit from activities in which she or he is engaged as a self-employed individual or as an employee is considered to be earned income.
Some or all countable income may be disregarded through deductions in some step of the eligibility determination process.
www.workworld.org /wwwebhelp/countable_income_food_stamps.htm   (334 words)

 : SI 00810.300 - Computing Countable Income - General - 01/11/2000   (Site not responding. Last check: 2007-10-22)
CI is the sum of a month's countable earned and countable unearned income, and includes the value of the one-third reduction (VTR) (see SI 00810.300B.4.
Countable earned income is the amount of earned income (SI 00810.015A.2.
Countable unearned income is the amount of unearned income (SI 00810.015A.3.
policy.ssa.gov /poms.nsf/lnx/0500810300   (203 words)

 Elder Law: Medicaid Countable Assets
Your pension fund is countable as an asset only to the extent that you have the right to remove the funds as a lump sum.
Like a pension, an annuity is countable only to the extent that you can remove the funds as a lump sum.
Such loan proceeds would be countable as cash assets if a loan has been taken out, unless the proceeds are spent on consumable or non-countable assets.
www.eldernet.com /lawcount.htm   (703 words)

 Countable and Uncountable Nouns   (Site not responding. Last check: 2007-10-22)
Countable nouns typically occur as discrete units that can be counted, whereas uncountable nouns, also called mass nouns, are typically substances or concepts.
Although it is often easy to distinguish between countable and uncountable nouns, the distinction is not always obvious for all nouns.
For this reason, you may find it useful to consult a specialized dictionary when you are not sure whether a particular noun is countable or uncountable or whether it has special meanings in one category or the other.
mit.imoat.net /handbook/n-c-unc.htm   (151 words)

 Number and Operations Session 2, Part B: The Size of Infinity
It is infinite, but it can be called countably infinite because you can put it in one-to-one correspondence with the counting numbers.
We know that rational numbers are countably infinite and that the real numbers are not (as you've seen in the video segment).
If, however, you add something to a countable set and get an "uncountable" set as a result, the second set must be uncountable as well.
www.learner.org /channel/courses/learningmath/number/session2/part_b   (694 words)

 [No title]   (Site not responding. Last check: 2007-10-22)
Here's an analogy that might help to show where your argument breaks down: Each semi-open interval (-oo, r) in the real line for any real number r is the union of all sets V of the form (-oo, x) for x a rational number less than r.
As for what you want to prove, namely that for countable topological spaces, first countability implies second countability, you want to be a bit more explicit in your proof.
[Each point has a countable local base (these bases may not be unique, so make sure you don't refer to any of them as "the local base at x" until you've fixed a choice of them in your proof), there are countably many points, a countable union of countable sets is countable,.
www.math.niu.edu /~rusin/known-math/01_incoming/1st_countable   (646 words)

 OWL at Purdue University: Count and NonCount Nouns (with Articles and Adjectives)
A countable noun always takes either the indefinite (a, an) or definite (the) article when it is singular.
The is sometimes used with uncountable nouns in the same way it is used with plural countable nouns, that is, to refer to a specific object, group, or idea.
These terms and their uses are summarized in the chart on our other countable and uncountable nouns page.
owl.english.purdue.edu /handouts/esl/eslcount2.html   (679 words)

 Countable Infinity
An alternate way to define countable is: if there is a way to enumerate the elements of a set, then the set has the same cardinality as N and is called countable.
Note that there is a difference between finite and countable, but we will often use the word countable to actually mean countable or finite (even though it is not proper).
Show that all polynomials (with integer coefficients) are countable by writing that set as a countable union of countable sets.
pirate.shu.edu /projects/reals/infinity/countble.html   (911 words)

 Retirement Benefits: Medicaid: Countable Assets
If the person's countable assets are less than $2,000, the individual will meet the resource requirements, even if the total of countable and non-countable assets combined is over the limit.
For married applicants, any asset that would be countable if the applicant owned it is also countable if the spouse owns it, and vice versa.
Mutual fund investments, for example, are countable, even if they are held solely in the name of the applicant's spouse.
www.eldernet.com /retiremt/assets.htm   (227 words)

 Mudd Math Fun Facts: How many Reals?
that the rational numbers are countable, meaning they have the same cardinality as the set of natural numbers.
If you have time show Cantor's diagonalization argument: if the reals were countable, we could use the 1-1 correspondence with the natural numbers to list them in order given by those natural numbers.
The theory of countable and uncountable sets came as a big surprise to the mathematical community in the late 1800's.
www.math.hmc.edu /funfacts/ffiles/30001.4.shtml   (332 words)

 Topological space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-22)
A space is (Click link for more info and facts about separable) separable if it has a (Click link for more info and facts about countable) countable dense subset.
A space is (Click link for more info and facts about second-countable) second-countable if it has a (Click link for more info and facts about countable) countable base for its topology.
A space is countably compact if every countable open cover has a finite subcover.
www.absoluteastronomy.com /encyclopedia/t/to/topological_space.htm   (3343 words)

 Countable and Uncountable Sets
A set is countable if it is finite, or it can be placed in 1-1 correspondence with the positive integers.
The nonnegative integers are countable, as shown by the bijection f(n) = n+1.
We just showed the polynomials in x are countable, and since these act as coefficients, the polynomials in x and y are countable.
www.mathreference.com /set-card,cable.html   (710 words)

 Probability - Open Encyclopedia   (Site not responding. Last check: 2007-10-22)
Probability distributions can also be specified via moments or the characteristic function, or in still other ways.
A distribution is called a discrete distribution if it is defined on a countable, discrete set, such as a subset of the integers.
A distribution is called a continuous distribution if it has a continuous distribution function, such as a polynomial or exponential function.
open-encyclopedia.com /Probability   (2628 words)

 EMail Msg <9305241234.AA14945@turing.pacss.binghamton.edu>
Even countably infinite sets cannot be implemented, although many theorems and proofs are simpler when they are stated without any finite upper bounds.
Therefore, I believe that talk of countable infinities in foundational studies is OK, but that any generalization to uncountable cardinalities is not only unnecessary and irrelevant, but potentially dangerous.
The second is to show, as was assumed in the proof, that the real numbers are not countable, that there is a way to find the first number for every set of real numbers.
www-ksl.stanford.edu /email-archives/interlingua.messages/338.html   (2409 words)

 Countable Earnings - SSDI (Title II)
For an explanation of the various ways in which the terms "Countable Earnings" and "Countable Income" are used in the Social Security programs, see: Countable Income and Countable Earnings Overview —SSI/DI.
The "countable income" used to calculate your SSI benefit amount is the sum of your "countable earned income" and your "countable unearned income".
This "countable earned income" is very different from the "countable earnings" used to determine SGA.
www.workworld.org /wwwebhelp/countable_earnings_di.htm   (366 words)

 Proposition: Rationals are Countable   (Site not responding. Last check: 2007-10-22)
But then each row is countable (in fact, finite), and there are countably many rows.
Hence, we have written the rationals as a countable union of countable sets, so that the rationals are indeed countable.
But if that set is countable, then the subset of rational numbers of that set must also be countable, because it is certainly not finite.
www.shu.edu /projects/reals/infinity/proofs/cntbl_q.html   (145 words)

 No Title
We shall go over some of it in class, and some of the problems are left as exercises.
Definition 2 A set X is countable if it is countably infinite or finite.
This is often phrased as saying the countable union of countable sets is countable.
www.rpi.edu /~piperb/analysis1/countable   (235 words)

 Math Forum - Ask Dr. Math
Date: 05/18/2002 at 12:44:29 From: Doctor Paul Subject: Re: Countability of Primes and Composites Well, we know that the natural numbers (another name for the positive integers) are countable.
Thus the set of primes and the set of composite numbers are both countable (since they are both subsets of the natural numbers).
It is not possible to take the union of an uncountable set A and any set B (either finite, inifinite countable or infinite uncountable) and produce a countable set.
mathforum.org /library/drmath/view/60781.html   (382 words)

 ESL: Countable and Nouncountable Nouns
As shown above, the noun "protein" as a countable noun usually requires the article a if its is singular.
In this case, we are referring to a specific protein--hemoglobin--so the definite article the is used.
: To classify a noun as either countable or non-countable, it is often necessary to determine the meaning of the noun in the context of the sentence.
bio-text.com /ESL2a.html   (606 words)

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