Basically just the definition: Any open cover of a compact set has a finite subcover. To get a compact set, take the closure of S; to get an open cover, consider the union of all balls of radius d around every point in S...What property of compactness did you need?
Tedjn said:That is pretty slick. Occasionally I have these bouts where I realize I'm in out of my depth and don't know enough in this case about metric spaces. I know that in Rn compactness isn't needed (at least not directly). But now I'm not so sure about arbitrary metric spaces.
VKint said:Basically just the definition: Any open cover of a compact set has a finite subcover. To get a compact set, take the closure of S; to get an open cover, consider the union of all balls of radius d around every point in S...
Tedjn said:What property of compactness did you need? Here is an idea that might nudge you in the right direction. Consider the special case of R, which is one-dimensional. Since S is bounded, it lies entirely between -M and M for some M. Certainly, if d >= 2M, we need just pick any point in S to be done. Every other point is within a distance 2M from that point. What if d = M? A first thought might be to choose the midpoint... but that might not be in S. How can we compensate?
prettymidget said:I don't know much topology, only the basic concepts and definitions such as limit point, open closed sets, boundary, closure, and such. The class is a multivariable analysis class, so only basic knowledge has been needed so far.
I actually asked my TA the question and the proof he gave was very long and something that didn't even look undergraduate level to me. If you've found a proof using basic concepts I would appreciate it greatly if you would post it. I haven't really been able to dent the problem.
I'm not sure. I thought about the problem a lot and I was completely convinced that the set S need be both bounded and closed for the rest to follow, so its limit points would be contained in the set.
The boundedness of a set in n space is the property of the set having a finite or limited extent in all directions. In other words, the set is contained within a finite region in n-dimensional space.
To prove the boundedness of a set in n space, we must show that the set is contained within a finite region and does not extend infinitely in any direction. This can be done by showing that the distance between any two points within the set is finite.
No, a set in n space can either be bounded or unbounded, but not both. A set is considered bounded if it is contained within a finite region, while an unbounded set extends infinitely in at least one direction.
The dimension of n space does not affect the boundedness of a set. A set can be bounded in any dimension, as long as it is contained within a finite region and does not extend infinitely in any direction.
The concept of boundedness of sets in n space is important in various fields such as physics, engineering, and computer science. It is used to analyze the behavior of physical systems, design efficient algorithms, and ensure the stability of numerical methods.