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Simple topological problem

  1. Sep 29, 2007 #1
    I'm really stuck on this simple problem: Let X be a topological vector space and U, V are open sets in X. Prove that U+V is open.

    It should be a direct consequence of the continuity of addition in topological vector spaces. But continuity states that the [tex]f^{-1}(V)[/tex] is open whenever V is open, but not the converse. It would work if I showed that adding a constant is a homeomorphism, but I don't think this is the way I should do it. Is there any more simple way, that I overlooked?
     
  2. jcsd
  3. Sep 29, 2007 #2
    In fact it is sufficient to assume the other one of the sets to be open. Suppose V is open. If you succeed in proving that for all vectors u the set

    [tex]
    u+V = \{u+v\;|\;v\in V\}
    [/tex]

    is open, then you are almost done.
     
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