Closed and bounded in relation to compact

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  • #1
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So this is more so a general question and not a specific problem.

What exactly is the diefference between closed and boundedness?

So the definition of closed is a set that contains its interior and boundary points, and the definition of bounded is if all the numbers say in a sequence are contained within some interval. But isn't that the same thing as being closed?
 

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  • #2
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[itex]\mathbb{R}[/itex] is closed but not bounded.
[itex][0,1)[/itex] is bounded but not closed.

Do these examples help?
 
  • #3
HallsofIvy
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It can be proved that any compact subspace of a metric space (you need the metric to define "bounded") is both closed and bounded. Any subset of the real numbers (or, more generally, Rn) that is both closed and bounded is compact. But in other spaces, such as the Rational numbers with the metric topology, that is not true. And, of course, you can have compact sets in non-metric spaces where "bounded" cannot be defined (though compact sets are still closed in any topology).
 
  • #4
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(though compact sets are still closed in any topology).
Compact sets are only closed in Hausdorff topologies.
 
  • #5
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[itex]\mathbb{R}[/itex] is closed but not bounded.
[itex][0,1)[/itex] is bounded but not closed.

Do these examples help?

So what your examples are saying that R has some finite value (though we can never find it) at which R will end, but it is not within an interval?

I see the concept in the second example though.
 
  • #6
pwsnafu
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So what your examples are saying that R has some finite value (though we can never find it) at which R will end, but it is not within an interval?
You need to brush up on your definitions, closed doesn't mean that at all.
 
  • #7
lavinia
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bounded means that distances can not exceed a bound. R is not bounded because there are points of arbitrarily large distance away from each other. [0,1) is bounded because no two points can get more than a distance of 1 away from each other.

on the real line closed means that every convergent sequence converges inside the set. So all of R must be closed since it is the whole set. But [0,1) is not closed because the sequence

1/2, 3/4, 7/8, 15/16 ... is inside the set but it converges to 1 which is outside the set.
 
  • #8
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bounded means that distances can not exceed a bound. R is not bounded because there are points of arbitrarily large distance away from each other. [0,1) is bounded because no two points can get more than a distance of 1 away from each other.

on the real line closed means that every convergent sequence converges inside the set. So all of R must be closed since it is the whole set. But [0,1) is not closed because the sequence

1/2, 3/4, 7/8, 15/16 ... is inside the set but it converges to 1 which is outside the set.


Ok. I understand now what it means to be closed, but bounded is still a little fuzzy. When it comes to the bound, is the bound something that we select in order for our aribitrary distance to be satisfied?
 
  • #9
HallsofIvy
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"Bounded" means there is an upper bound on distances between points.
 
  • #10
lavinia
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Ok. I understand now what it means to be closed, but bounded is still a little fuzzy. When it comes to the bound, is the bound something that we select in order for our aribitrary distance to be satisfied?
there is an idea of a least upper bound which is the smallest number that bounds the distances between pairs of points. But larger numbers are also bounds.
 

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