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As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as answers so they may be voted up and down with the rest.

This is a community-wiki question: One colloquial statement and its mathematical meaning per answer please!

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    $\begingroup$ Recent answers suggest this question is getting a bit long in the tooth. $\endgroup$ – S. Carnahan Jul 6 '12 at 5:38

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A drunk man will find his way home, but a drunk bird may get lost forever.

This encodes the fact that a 2-dimensional random walk is recurrent (appropriately defined for either the discrete or continuous case) whereas in higher dimensions random walks are not. More details can be found for instance in this enjoyable blog post by Michael Lugo.

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    $\begingroup$ This particular saying, by the way, is usually attributed to Shizuo Kakutani. (I don't want anybody thinking I came up with it!) $\endgroup$ – Michael Lugo Oct 31 '09 at 17:16
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    $\begingroup$ And this is the reason that birds do not drink alcohol $\endgroup$ – user49822 Sep 4 '14 at 10:37
  • $\begingroup$ I feel it should be a drunk spaceship, since birds fly at most over a thickened 2-sphere, and realistically over just a small patch. $\endgroup$ – isomorphismes Apr 30 '15 at 20:35
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"Can you hear the shape of a drum?"

This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The answer, by the way, is "no", at least if one allows the domain to have corners.)

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An anagram for "Banach Tarski" is "Banach Tarski Banach Tarski"

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    $\begingroup$ I don't think this is what Armin Straub was looking for, but it certainly made me laugh. ^_^ $\endgroup$ – Vectornaut Nov 17 '09 at 19:10
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    $\begingroup$ I'd like to know the original source of this. I put it on a T-shirt. thenerdiestshirts.com/site/math-t-shirts-banach-tarski $\endgroup$ – Douglas Zare Jul 4 '12 at 21:49
  • $\begingroup$ Well, anyway it should be Hausdorff. $\endgroup$ – Anton Petrunin Apr 21 '13 at 22:12
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    $\begingroup$ If only King Solomon had known about the Banach-Tarski paradox. Instead of cutting the baby into two pieces, he could have suggested 5 pieces and made both mothers happy. $\endgroup$ – none Mar 2 at 8:13
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Take a map of wherever you are and lay it on the ground. There will be exactly one point on the map that is directly above the point it represents on the ground.

This refers to Banach's fixed point theorem.

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    $\begingroup$ This should also follow from Brouwer's fixed-point theorem, right? If so, it's the best statement of Brouwer's fixed-point theorem I've ever heard, and I'll definitely be using it in the future! $\endgroup$ – Vectornaut Nov 17 '09 at 19:02
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    $\begingroup$ You need Banach's fixed point for the uniqueness, but the existence follows Brouwer's fixed point. $\endgroup$ – Josiah Sugarman Dec 10 '09 at 15:59
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    $\begingroup$ i actually did this in class to demonstrate the fixed point theorem. there was a satisfying gasp when I picked up my lecture notes and scrunched them :) $\endgroup$ – Suresh Venkat Dec 10 '09 at 20:36
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    $\begingroup$ With this formulation one may say that even the proof somewhat made its way thorugh and reached literature. In Borges "Partial Enchantments of the Quixote" a story contained in Other Inquisitions, an apocryphal quote states that in a perfect map, a copy of the map should be contained, and such copy would contain another copy of the map and so on at infinity, which is basically the proof of the Theorem, if you add to it existence of the limit... $\endgroup$ – Nicola Ciccoli Jul 5 '12 at 7:24
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Complete disorder is impossible.

This is the standard way of summing up Ramsey theory in a succinct sentence (according to that Wikipedia article, the above quote is due to Motzkin).

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    $\begingroup$ By the way, Motzkin's remark, which is in a paper in J. Comb. Theory 3:244-252 (1967) made an interesting comparison between disorder in physics and combinatorics: "Whereas the entropy theorems of probability and mathematical physics imply that, in a large universe, disorder is probable, certain combinatorial theorems imply that complete disorder is impossible." $\endgroup$ – John Stillwell Jan 31 '10 at 6:04
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It's not possible to comb a hedgehog.

Or, alternatively:

You can't comb the hair on a coconut.

Both statements are referring to the fact that every continuous tangent vector field on the 2-sphere has to vanish at some point. That's the well known Hairy ball theorem.

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    $\begingroup$ Armin, you obviously travel in more poetic circles than I! I'd always just heard “you can't comb a hairy ball”. $\endgroup$ – LSpice Feb 24 '10 at 0:06
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    $\begingroup$ Of course one can comb a hedgehog, but not without a bald point! $\endgroup$ – Konrad Waldorf Jul 5 '12 at 8:37
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    $\begingroup$ Those who comb hedgehogs for a living know that they have bald bellies: goo.gl/E1HWh $\endgroup$ – Theo Johnson-Freyd Aug 25 '12 at 13:25
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"Space tells matter how to move; matter tells space how to curve."

This is the quintessential colloquial expression of the Einstein field equations that govern general relativity.

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  • $\begingroup$ Good one\!!!!!! $\endgroup$ – Felix Goldberg Jul 5 '12 at 14:35
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    $\begingroup$ Punchy as that is, I regret that it was not bequeathed to us in the form "Space tells matter where to go; matter tells space to get bent." $\endgroup$ – LSpice Feb 14 '16 at 14:42
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"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard

I guess he meant that often the best proof of a theorem about real numbers uses complex analysis.

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    $\begingroup$ No, he meant something even better. A real integral is often best found by a path through the complex plane. $\endgroup$ – Colin McLarty Feb 26 '14 at 21:04
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All primes are odd except 2, which is the oddest of all.

This has been discussed before. The quote is from "Concrete Mathematics," but it's a rephrasal of one of J.H. Conway in "The book of numbers."

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    $\begingroup$ Concrete Mathematics (1989, 2nd ed. 1994) was published before The book of numbers (1996). $\endgroup$ – Michael Lugo Nov 3 '09 at 23:21
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    $\begingroup$ I heard it from John Thompson in 1960. As well as an other inadvertent quote of his when he was speaking to an audience at a catholic school about variants of the Cantor set--" there's nothing sacred about the number 3". $\endgroup$ – paul Monsky Feb 14 '16 at 23:46
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"A projective module is the splittin' image of a free module."

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  • $\begingroup$ Ha ha :) made me laugh, or at least grin wryly. $\endgroup$ – David Roberts Apr 19 '11 at 5:26
  • $\begingroup$ Love it! (and love it, to fill out more characters) $\endgroup$ – Todd Trimble May 8 '11 at 20:34
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    $\begingroup$ "X is the spitten image of Y" is an informal phrase meaning "X looks just like Y". "X is the spitting image of Y" is an entirely equivalent phrase; in some dialects, the pronunciation is even the same. As far as I know, neither is the "right spelling"; the phrase rarely appears in written English. $\endgroup$ – jasomill May 10 '11 at 10:48
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    $\begingroup$ @Rasmus: The English colloquialism uses "spit", so that "split" is a pun. See english.stackexchange.com/questions/8509. $\endgroup$ – Theo Johnson-Freyd Aug 25 '12 at 13:28
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    $\begingroup$ It may be that the original is "spit and image". $\endgroup$ – Tom Goodwillie Jun 9 '15 at 11:20
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"If you are walking between two policemen going to the same station, you will end up there, too."

This encodes the familiar Squeeze Theorem: If $a_n,b_n, c_n$ are sequences of real numbers such that $a_n \leq b_n \leq c_n$ and $\lim_{n \to \infty}a_n =\lim_{n \to \infty} c_n$, then $\lim_{n \to \infty}a_n =\lim_{n \to \infty} c_n= \lim_{n \to \infty}b_n$.

I am not sure whether it counts as "serious mathematics", but this is how I learned it as a high school student in Communist-ruled Poland.

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  • $\begingroup$ In France it is called the "policemen theorem" too. $\endgroup$ – Thomas Richard Apr 19 '11 at 7:12
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    $\begingroup$ I learned it in an Israeli high school from a Russian teacher and he called it two policemen and a drunk. So the drunk is between the two policmen who are going to the station. $\endgroup$ – Yiftach Barnea Apr 19 '11 at 7:19
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    $\begingroup$ And then there is the running joke of calling it the "three policemen" theorem because the drunk is a policeman as well. $\endgroup$ – darij grinberg Apr 19 '11 at 8:29
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    $\begingroup$ In Italy too (or rather the "Carabinieri" theorem, soldiers of a police force commonly regarded as ridiculous) $\endgroup$ – Filippo Alberto Edoardo Jul 5 '12 at 3:18
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    $\begingroup$ Of course, in most practical applications one policeman is 0 and the other tends to 0. $\endgroup$ – Omer Aug 6 '12 at 0:14
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"If it walks like a sphere and it quacks like a sphere then it is a sphere."

A professor at my university explained the Poincare Conjecture to his 1st semester abstract algebra students this way. I think it is a great explanation!

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There is a pair of antipodal points on the surface of the Earth at equal temperature and equal pressure.

The Borsuk-Ulam theorem for n=2. Suppose $f: \mathbb{S}^n \rightarrow \mathbb{R}^n$ is a continuous map. Then $\exists x: f(x) = f(-x)$.

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    $\begingroup$ While the pressure field should be continuous, I can see no reason why the temperature field have to be continuous? $\endgroup$ – kjetil b halvorsen Aug 26 '12 at 4:32
  • $\begingroup$ @kjetilbhalvorsen Indeed, near-zero-Kelvin labs on Earth mean it's probably not. $\endgroup$ – isomorphismes May 1 '15 at 14:39
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If n people are in an elevator and n+1 buttons are pushed, there is at least one pigeon brain in the elevator.

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    $\begingroup$ Who you calling pigeon brain? youtube.com/watch?v=mDntbGRPeEU :-) $\endgroup$ – Willie Wong Apr 19 '11 at 0:47
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    $\begingroup$ "If there are fewer pigeons than holes, there must be a pigeon with at least two holes in it." $\endgroup$ – Akiva Weinberger Sep 2 '15 at 20:58
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Period three implies chaos.

This summarizes a paper (of the same name) by Li and Yorke. The full statement of the main theorem is that if a continuous transformation of an interval has a point whose orbit has length three, then there exist points whose orbits are completely chaotic (in addition to points with orbits of every other possible finite length).

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A brain tangled enough to tackle itself must be too tangled to tackle.

A neat formulation of Goedel's Incompleteness Theorem. From the novel "Galatea 2.2" by R.Powers.

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    $\begingroup$ Well, it takes two to tangle. $\endgroup$ – Benjamin Dickman Aug 25 '12 at 9:40
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Two plus two is four. You can prove that two plus two is four. You can prove that you can prove that two plus two is four. And you can prove that you can prove that you can prove that two plus two is four, and so on.

Two plus two is not five. You can prove that two plus two is not five. You can't prove that two plus two is five, or else math is a lot of bunk. But, if math is not a lot of bunk, you can't prove that you can't prove that two plus two is five.

(Shortened from Gödel's Second Incompleteness Theorem Explained in Words of One Syllable by George Boolos, Mind, Vol. 103, January 1994, pp. 1 - 3.)

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Odd is better, but three is best.

This refers to the fact that "sound" propagation in arbitrary dimensions (using the wave equation on an initial pulse) is only possible in odd-dimensional spaces (in even dimensions there's an ever-lasting echo), and that dispersion-free propagation only happens for dimension three.

See this paper where I saw the quote: part one, part two, although the fact itself seems to be known for ages (I have no access but an old paper by Balazs seems relevant too), possibly this even goes back to Petrovskii according to an interview of Arnold (see p434).

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    $\begingroup$ This is also known as the "strong Huygens principle". $\endgroup$ – Terry Tao Nov 1 '09 at 21:29
  • $\begingroup$ And one of the better tongue-in-cheek titles of a paper I've seen: "A Simple Proof that the World is Three-Dimensional". $\endgroup$ – K. Henriksen Apr 9 at 3:23
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"Chewy on the outside, crunchy on the inside."

The way that Dick Canary would characterize hyperbolic 3-manifolds with nonempty (and nonrigid) conformal boundary.

"Crunchy on the outside, chewy on the inside."

Description of Rich Schwartz's complex hyperbolic manifold with conformal boundary a real hyperbolic manifold.

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Fields medalist René Thom famously wrote:

Ce qui limite le vrai n'est pas le faux, mais l'insignifiant. (Approx translation: "What limits truth is not wrongfulness, it's meaninglessness.")

This refers to the basic mathematical issue that one must only consider well-formed formulas rather than arbitrary ones.

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    $\begingroup$ Is that really what he meant? That seems like a pretty narrow interpretation. $\endgroup$ – Darsh Ranjan Nov 2 '09 at 2:48
  • $\begingroup$ Nice, but I'm not convinced by the translation. Here's my take: "Truth is not bounded by falsity, but by the insignificant", which I imagine means something like: "A true statement is worthless if its scope is very limited." (I'd never heard this statement and have no idea what Thom's intentions were, but being French I can perhaps help point to the way out of meaninglessness) $\endgroup$ – PatrickT Apr 18 at 7:58
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A city is compact if it can be patrolled by finitely many nearsighted police officers.

I believe this is due to Peter Lax. Of course one must take some care with quantifiers to make this a correct definition, but I think it captures the spirit nicely.

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    $\begingroup$ I think this is a dangerous metaphor: it encourages the misconception that "a space is compact iff it has a finite open cover". $\endgroup$ – Rasmus May 10 '11 at 8:04
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    $\begingroup$ This is attributed to Weyl in Wilson Sutherland's text Introduction to Metric and Topological Spaces (section 5.2). Sutherland gives it as "If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen". Exercise 5.10.15 asks you to make precise, and discuss the accuracy of, Weyl's statement. $\endgroup$ – Tom Leinster Aug 5 '12 at 21:40
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    $\begingroup$ A city is compact if you can fire all except finite number of policemen patrolling the city $\endgroup$ – Ostap Chervak Jan 31 '13 at 10:21
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    $\begingroup$ @TomLeinster The original may well be this, from Weyl’s Harmonics on homogeneous manifolds, p. 489: «A “finite” country can be watched by a finite number of policemen, however small the radius of action of the single policeman may be!» $\endgroup$ – Francois Ziegler Jun 28 '18 at 17:30
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Black holes have no hair.

This no-hair theorem "postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum".

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    $\begingroup$ This is of course true only in 4 dimensions. Higher-dimensional black holes can be hairy. $\endgroup$ – José Figueroa-O'Farrill Nov 2 '09 at 5:54
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After stirring a cup of coffee, at least one point will end up in the exact same position as it was before.

Being an approximate statement of Brouwer's fixed point theorem.

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    $\begingroup$ I've always had issues with this one since I'm not convinced the "stirring" function would be continuous... $\endgroup$ – Elisha Peterson Nov 1 '09 at 20:11
  • $\begingroup$ Indeed, if you place the spoon vertically in the middle of the coffee mug down to the bottom, the coffee will cease to be contractible. $\endgroup$ – Michael Greinecker Jul 2 '17 at 10:47
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Time Average is the Space Average

which is the the Ergodic Theorem.

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    $\begingroup$ It seems like a reasonable way of summarizing it catchily, but I'd prefer a little extra context -- as stated it reads like it could be something from Time Cube... $\endgroup$ – Harrison Brown Nov 1 '09 at 15:43
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Trust, but verify.

this is the defn of NP, of course, but for a certain generation, it has Reaganesque overtones.

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    $\begingroup$ According to the Wikipedia page on this catchphrase (yes, Wikipedia has a "trust, but verify" page), it originates in a Russian proverb, as Reagan himself introduced it. $\endgroup$ – Noam D. Elkies Aug 1 '11 at 4:34
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All diagrams commute.

The actual theorem is Mac Lane's Coherence Theorem or any of a number of other coherence theorems that guarantee that all category diagrams built from certain elements commute.

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Local is global

The very first thing I heard about sheaves, from another graduate student.

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    $\begingroup$ A variant (with different mathematical overtones): "Think globally, act locally". I had a button with this on it as an undergraduate, from a very non-mathematical source. $\endgroup$ – Ravi Vakil Jan 31 '10 at 5:38
  • $\begingroup$ This reminds me of a Gogol Bordello song... "Think locally ______" $\endgroup$ – Sean Tilson Jul 5 '12 at 21:10
  • $\begingroup$ The same is true for convex optimization.... $\endgroup$ – Suvrit Aug 26 '12 at 16:47
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Another one: not so much a catchphrase, but a nifty interpretation of a theorem:

"Suppose a human is walking a dog on the leash and they encounter a lamp post. Then, if the leash is kept short enough, the human and the dog wind around the post the same number of times."

I learned this interpretation of Rouche's Theorem from the textbook in complex analysis by Saff and Snider. They include pictures, too.

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    $\begingroup$ By “the leash is kept short enough”, do you mean it can't be flung over the lamp? $\endgroup$ – Zsbán Ambrus Jul 5 '12 at 21:31
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If you save a penny a day, eventually you will become a millionaire. This is a loose paraphrase of the Archimedean Property of Archimedean Ordered Fields:

If $\epsilon \gt 0$ and if $M \gt 0$, then $\exists$ an $n\in\mathbb{N}$ such that $n\epsilon\gt M$.

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  • $\begingroup$ This property has another paraphrase: even taking very small steps one can walk a very long way (known in Polish as "twierdzenie o malych piechurach", i.e, "theorem on little walkers"). Sounds like Comrade Mao... $\endgroup$ – Margaret Friedland Jul 5 '12 at 14:46
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Being compact is the next best thing to being finite.

As seen for example by the fact that the (uniform) limit of a sequence of continuous real-valued functions on a compact space is continuous. Or the even more down to earth example: The extreme value theorem.

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    $\begingroup$ I've heard a minor variation: "Compact is the new finite." $\endgroup$ – S. Carnahan Aug 6 '12 at 0:47

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