Closed Subset Addition in Metric Spaces: Real Analysis Homework Help

In summary, the conversation discusses the closure of a set E+F, where E and F are closed and non-empty subsets of a metric space with a given distance formula. The question asks if E+F is necessarily closed. The attempt at a solution provides two examples, one involving a compact set and the other involving the integers and irrational multiples of sqrt(2).
  • #1
Mr_Physics
6
0

Homework Statement



Let E, F be two closed and non-empty subsets of R, where R is seen as a metric space with teh distance d(a,b)=|a-b| for a,b ϵ R.

Suppose E + F := { e+f |e ϵ E, f ϵ F}. Is is true that E+F has to be closed?

Homework Equations





The Attempt at a Solution



I'm not sure how to start this one.
 
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  • #2
This is not an easy one, by far!

But consider:

[tex]A=\{...,-4,-3,-2,-1\}[/tex]

and

[tex]B=\{1+1/2,1+1/2+1/3,1+1/2+1/3+1/4,...\}[/tex]

Note that E+F is closed if one of E or F is compact...
 
  • #3
micromass said:
This is not an easy one, by far!

But consider:

[tex]A=\{...,-4,-3,-2,-1\}[/tex]

and

[tex]B=\{1+1/2,1+1/2+1/3,1+1/2+1/3+1/4,...\}[/tex]

Note that E+F is closed if one of E or F is compact...

I'll agree it's not easy. It does take some head scratching. Here's another one to think about on a different line. Take A=Z (the integers) and B=Z*sqrt(2). micromass's suggestion is really pretty clever though once you figure it out.
 
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FAQ: Closed Subset Addition in Metric Spaces: Real Analysis Homework Help

1. What is "Closed Subset Addition" in metric spaces?

"Closed Subset Addition" in metric spaces is a concept in real analysis that deals with the properties of closed subsets in a metric space. It involves studying the behavior of closed subsets under addition, and how this operation affects their properties.

2. Why is "Closed Subset Addition" important in real analysis?

By understanding the properties of closed subsets and how they behave under addition, we can gain a better understanding of the structure of metric spaces. This can help us in proving theorems and solving problems in analysis.

3. What are some key properties of "Closed Subset Addition" in metric spaces?

Some key properties of "Closed Subset Addition" include closure under addition, meaning that the sum of two closed subsets is also a closed subset; associativity of addition; and the existence of a neutral element (the empty set) under addition.

4. How is "Closed Subset Addition" related to other concepts in real analysis?

"Closed Subset Addition" is closely related to the concepts of open subsets and compactness in metric spaces. It also has connections to the notions of convergence and continuity in analysis.

5. Can you give an example of "Closed Subset Addition" in action?

One example of "Closed Subset Addition" is in proving the closure of a set. If we have a set A that is closed under addition, we can show that the closure of A is also closed under addition by proving that the limit of any convergent sequence in the closure of A is also in the closure of A.

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