A misunderstanding of compact sets

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I am trying to understand the definition of compact sets (as given by Rudin) and am having a hard time with one issue. If a finite collection of open sets "covers" a set, then the set is said to be compact. The set of all reals is not compact. But we have for example:

C1 = (-∞, 0)
C2 = (0, +∞)
C3 = (-1, 1)

ℝ ⊆ C1 ∪ C2 ∪ C3

The first thing I can think of as a problem is that the endpoints of two of the sets are infinities. But when I read the definition of an open set, that doesn't seem to pose a problem. What am I missing?
 

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  • #3
Stephen Tashi
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If a finite collection of open sets "covers" a set, then the set is said to be compact.
That isn't the definition of a compact set.

(Each open set covers itself and so is covered by a finite collection of open sets. We don't want a definition of "compact" that makes each open set a compact set.)
 
  • #4
lurflurf
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The definition does not say a set is compact if it has a finite open cover. The definition says a set is compact if every open cover has a finite subcover. In you example indeed R has the finite open cover
C1 = (-∞, 0)
C2 = (0, +∞)
C3 = (-1, 1)

However R has other open covers such as
Ck=(k-10^-10,k+10^-10)
k an integer

This open cover has no finite subcover as if we remove (k-10^-10,k+10^-10) we have lost real numbers.
 
  • #5
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The way I like to phrase the definition is: given an infinite open cover, you can finite it.
 
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Okay, that clarifies it, thanks guys! Sorry, I'm not very experienced in these types of more rigorous approaches to math yet and sometimes fail to see/remember certain parts of definitions which turn out to be crucial.
 
  • #7
HallsofIvy
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The way I like to phrase the definition is: given an infinite open cover, you can finite it.
I don't believe I have ever seen "finite" used as a verb before!
 
  • #8
Stephen Tashi
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certain parts of definitions which turn out to be crucial.
Those parts often involve the quantifiers "for each" and "there exists", the order in which they appear being important (for example, "for each ....there exists...such that ." versus "there exists .... such that for each....").

I find "for each" a clearer phrase than the ambiguous "for all", even though the symbol [itex] \forall [/itex] is named "forall" in LaTex.
 
  • #9
Fredrik
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I think the quantifier should be called "for all", but when I type out a statement in plain English, I will sometimes write "for each". Example:

For each positive real number x, there's a unique positive real number y such that ##y^2=x##.​

If we write "for all" here, English grammar says that we should change "number" to "numbers", and then the entire sentence gets weird.

But sometimes "for all" sounds better.

For all positive real numbers x, x has a square root. For all real numbers x, we have ##x^2>0##.​

"For each" is also very nice in definitions.

For each positive integer n, we define n! by n!=n(n-1).​
 
  • #10
Stephen Tashi
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The ambiguity of "for all" that I don't like is illustrated by the contrast between

"For all real numbers x, there exists a number y such that y > x"

versus

"For all real numbers x, there exists a number y such that x + y = x"
 
  • #11
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The ambiguity of "for all" that I don't like is illustrated by the contrast between

"For all real numbers x, there exists a number y such that y > x"

versus

"For all real numbers x, there exists a number y such that x + y = x"
Are you afraid that the second example might be misconstrued to mean the same thing as, "There exists a number y such that, for all real numbers x, x+y=x," or do you think the second example is actually intended to mean the same thing as what I have written?

Or to put it another way, is it possible that the ambiguity that you are experiencing is the result of not being careful with the order of the quantifiers as opposed to the meaning of "for all"?
 
  • #12
Fredrik
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The ambiguity of "for all" that I don't like is illustrated by the contrast between

"For all real numbers x, there exists a number y such that y > x"

versus

"For all real numbers x, there exists a number y such that x + y = x"
I don't understand. Why would it matter if the property satisfied by the dummy variable y is y>x or x+y=x?
 
  • #13
Fredrik
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On second thought, I'm not sure I think that "for each" is ever really better. I think I just let myself get confused by some plain English. When a sentence starts with "for all x", then the statement that follows is a true statement about the set represented by the variable x, regardless of what set that is. Since a variable can only represent one set at a time, there's no doubt that the rest of the sentence should be talking about x as if it's only one set (because it is...we're just making a statement that's true regardless of what set that is). For example, "for all positive real numbers x, x has a square root" is perfectly fine, even though the word "all" usually requires us to use the plural form, as in "all positive real numbers have square roots".
 

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