# What are the most overloaded words in mathematics? [closed]

This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or definitions. ("Adjective" and "adjective noun" count as the same adjective.) People should edit previous answers as appropriate.

(This is mostly just for fun, but I'm also curious if there have been successful attempts to rename concepts that involve overused words.)

Edit: I may have been slightly unclear about the intent of this question.

• When I say "overused" I don't mean "used too often," I mean "used in too many different ways." So I'll change the title of the question to reflect this.
• Different concepts named after the same mathematician, while potentially confusing, are understandable.
• I mostly had in mind adjectives that get recycled in different disciplines of mathematics. Different uses of the same noun tend to be less confusing, e.g. the example of "space" below. I think it's good to be intentionally vague about what we consider a "space."
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I'm -1'ing this because I think it's a rather uninteresting question. –  Kevin H. Lin Dec 1 '09 at 12:47
I'm +1'ing it because the answers are useful. –  Greg Kuperberg Dec 1 '09 at 18:54
I don't like the question because it invites too many bad answers. I (-1)ed a bunch of answers for the following reason. Just because an adjective is used with many different nouns or a word is used frequently doesn't mean that it's overloaded; the use may be perfectly consistent. I realize the the question was edited, so this is a slightly unfair criticism. –  Anton Geraschenko Dec 14 '09 at 16:55
I'm voting to close; I didn't really see the point of this when it was first posted, and I still don't really see the point now. –  Kevin H. Lin Apr 20 '10 at 22:27

## closed as no longer relevant by Harry Gindi, Noah Snyder, Qiaochu Yuan, François G. Dorais♦, Scott Morrison♦Apr 30 '10 at 4:56

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Fundamental theorem:

* Fundamental theorem of algebra
* Fundamental theorem of arithmetic
* Fundamental theorem of calculus
* Fundamental theorem of curves
* Fundamental theorem of cyclic groups
* Fundamental theorem of surfaces
* Fundamental theorem of finitely generated abelian groups
* Fundamental theorem of Galois theory
* Fundamental theorem on homomorphisms
* Fundamental theorem of linear algebra
* Fundamental theorem of projective geometry
* Fundamental theorem of Riemannian geometry
* Fundamental theorem of stochastic calculus
* Fundamental theorem of vector analysis
* Fundamental theorem of linear programming

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Isn't this a bit silly? Using the same argument, you could say that 'theorem' is even more overloaded... –  Mariano Suárez-Alvarez Apr 21 '10 at 7:37
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-ary

-ary, as in $k$-ary numeral $s$, refers to the number $k$ of values in the domain $K = \lbrace 0, 1, \ldots, k-1 \rbrace$ that affords the basis of numeration.

-ary, as in $k$-ary relation $L$, refers to the number of domains $X_1, \ldots, X_k$ for which $L \subseteq X_1 \times \ldots \times X_k$.

-ary, as in $k$-ary operation $f$, refers to the number of domains in the domain of the function $f : X_1 \times \ldots \times X_k \to Y$, the rubric being, "a $k$-ary operation is a $(k+1)$-ary relation".

Some writers use Greek roots and the Greek suffix "-adic" for the number of domains in a relation, hence medadic, monadic, dyadic, triadic for relations of 0, 1, 2, 3 places, respectively. This usage actually has a degree of historical precedence and it can serve to sidestep conflicts with the domainance of binary numerals in our modern world, but of course the wrinkle but moves to other domains where writers are adicted to other habits.

NB. All puns are intended.

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## Elementary

Sense 1 = basic, simple, concerning the elements ("first steps") of a subject.

• elementary fact
• elementary introduction
• elementary problem
• elementary proof

Sense 2 = treating mathematical objects as elements of a collection.

• elementary number theory
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Space

Affine space

Banach space

Cauchy space

Euclidean space

Function space

Hardy space

Hilbert space

Inner product space

Kolmogorov space

Krein space

Klein space

Pontrjagin space

Lp space

Measure space

Metric space

Minkowski space

Normed vector space

Locally convex space

Linear topological space

F-space

Frechet space

Nuclear space

Operator space

Affine space

Projective space

Polish space

Quotient space

Sobolev space

Topological space

Uniform space

Vector space

Harmonic spaces

Conformal space

Complex analytic space

Affinely connected space

Algebraic space

Symplectic space

Measurable space

Measure space

Probability space

Riemann space

Lorentzian space

And so on.

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I guess I should've been clearer. I don't find repeated use of the same noun confusing, and the question is mostly about adjectives. –  Qiaochu Yuan Dec 1 '09 at 17:50
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