Is the Sum of Two Closed Sets in R^n Always Closed?

In summary, the question is whether the sum of two closed sets in R^n is always closed. The conversation considers the possibility that this may not be true, and explores the idea of using sequences to prove or disprove this statement. The idea of counterexamples is suggested, and it is noted that the statement is true when one of the sets is finite. The conversation ends with the suggestion to continue exploring examples where both sets are infinite.
  • #1
teacher2love
2
0
Let A, B in R^n be closed sets. Does A+B = {x+y| x in A and y in B} have to be closed?

Here is what I've tried. Let x be in A^c and y in B^c which are both open since A & B are closed. So for each x in A^c there exists epsilon(a)>0 s.t. x in D(x, epsilon(a) is subset of A^c. For each y in B^c there exists epsilon(b)>0 s.t. y in D(y, epsilon(b)) is a subset of B^c.

Can I add the two together to get x+y in (A+B)^c to show that there exist epsilon > 0 s.t. D(x+y, epsilon) is in (A+B)^c. Thus (A+B)^c is open => (A+B) is closed.

Thanks for your help.
 
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  • #2
It is often much easier to work with sequences when you can!

A set is closed if the limit of every converging sequence of elements of that set lies in the set.
 
  • #3
teacher2love said:
Can I add the two together to get x+y in (A+B)^c to show that there exist epsilon > 0 s.t. D(x+y, epsilon) is in (A+B)^c. Thus (A+B)^c is open => (A+B) is closed.

It's not true that x+y will be in (A+B)^c. (almost any example you can think of will show this, but to take a simple one, let A=B={0}, x=-y=1).

A good strategy on these types of problems (where you're not sure if the given statement is true or false) is to start by trying to find counterexamples. If you find one, you're done, and if not, try to see what's preventing you from finding one.

For example, you might notice that you can't find any counterexamples when one of the sets is finite. Well, this is just because, as is easy to prove, A+B is closed when one of A or B is finite. So you can continue, now looking only at examples where both A and B are infinite.
 

Related to Is the Sum of Two Closed Sets in R^n Always Closed?

1. What is a closed set in topology?

A closed set in topology is a set that contains all of its limit points. In other words, for every convergent sequence in the set, the limit of that sequence is also in the set.

2. How do you determine if a set is closed in topology?

To determine if a set is closed in topology, you can use one of several equivalent definitions. One approach is to check if the complement of the set is open. If the complement is open, then the set is closed. Another approach is to check if the set contains all of its limit points.

3. What is the difference between a closed set and a compact set?

A closed set is a set that contains all of its limit points, while a compact set is a set that is both closed and bounded. Not all closed sets are compact, but all compact sets are closed.

4. Can a set be open and closed at the same time in topology?

Yes, in some topological spaces, a set can be both open and closed. These are known as clopen sets. In other topological spaces, there may be no clopen sets, or only the empty set and the whole space are clopen.

5. How is the closed set problem related to the Heine-Borel theorem?

The closed set problem is related to the Heine-Borel theorem in that the Heine-Borel theorem is a necessary and sufficient condition for a set to be compact in Euclidean spaces. This theorem helps to solve the closed set problem by providing a criterion for determining when a set is compact.

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