How do closed, bounded, and compact concepts relate in metric spaces?

In summary, the concepts of bounded, closed, and compact sets in a metric space are closely related. A set is bounded but not closed if it has a finite upper or lower bound but does not include its boundary points. A set is closed but not bounded or compact if it includes all of its boundary points but does not have a finite upper or lower bound. In R^n, a set is compact if and only if it is both closed and bounded. In a metric space, compactness is equivalent to completeness and total boundedness. In R^n, closed sets are equivalent to complete sets, and bounded sets are equivalent to totally bounded sets. The connection between these concepts is that a compact metric space is one in which every sequence has a
  • #1
Ratzinger
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Could someone explain me how these three concepts hang together?

(When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)
 
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  • #2
Examples (real line with usual topology).

Bounded not closed: 0<x<1
Closed not bounded or compact: 0<=x<oo.
 
  • #3
in R^n, compact is equivalent to closed and bounded, so a closed set is bounded iff compact, and a bounded set is closed iff compact.

in a metric space, compact is equivalent to complete and totally bounded.

in R^n which is itself complete, closed is equivalent to complete, and since every bounded set in R^n has comoact closure, bounded is equivalent to totally bounded.

a totally bolunded set is one in which everys equence ahs a cauchy subsequence, and a completes et one in which everyu cauchy sequence converges.

the connection is that a compact metric space is one in which every sequence has a convergent subsequence. (i think. it has been a long time since i taught this course.)
 

1. What does it mean for a set to be closed?

A set is considered closed if it contains all of its boundary points. In other words, a set is closed if it contains all of its limit points.

2. How is a set bounded?

A set is bounded if there exists a finite number M such that the absolute value of every element in the set is less than or equal to M. This means that the set is contained within a finite interval.

3. What is the difference between closed and compact?

Closed sets and compact sets are related but not the same. A set is closed if it contains all of its limit points, while a set is compact if it is both closed and bounded. In other words, every compact set is closed, but not every closed set is compact.

4. How can I determine if a set is compact?

There are several ways to determine if a set is compact. One method is to show that the set is both closed and bounded. Another method is to show that every open cover of the set has a finite subcover.

5. Why is compactness an important property in mathematics?

Compactness is important because it allows us to make generalizations about sets and functions. For example, if a set is compact, then we know that it is both closed and bounded, which can help us make conclusions about the set's behavior. Compactness is also useful in proving the existence of solutions to certain problems.

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