Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Topology closed sets problem

  1. Mar 5, 2008 #1
    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.
  2. jcsd
  3. Mar 5, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
  4. Mar 5, 2008 #3


    User Avatar
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook