- #1
AdrianZ
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Homework Statement
Suppose A is a compact set and B is a closed subset of Rk. then A+B is closed in Rk. show that A+B for two closed sets is not necessarily closed by a counter-example.
well, since A is a compact set and there's a theorem in Rudin's mathematical analysis chapter 2 stating that compact sets of metric spaces are closed, we can conclude that A is closed. that means every limit point of A is in A. B is closed by assumption, and that means every limit point of B is in B. now here is where I get confused. is A in Rk? I think it should be because if not then how A+B is defined? I want to define A+B as the set {a+b: A,B are in Rk} but the problem has ambiguity and doesn't precisely say what A is. Do we need to know what A is? or we can solve it without knowing what A is? It's easy to solve it if A is in Rk, but I don't know what I should do if A is not in Rk.