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The following is from Mariano's comments on my earlier question

  1. In a topological vector space, why is the following true:

    if a neighborhood U of zero contains a scaled copy of the whole space, then it is in fact the whole space.

    Is "a neighborhood U of zero contains a scaled copy of the whole space" the same as "a scaled copy of a neighborhood U of zero is the whole space"?

    I have thought about this for a while but don't know why.

  2. In a vector space, is it true that if a subset U of zero contains a scaled copy of the whole space, then it is in fact the whole space? I think it is not true when the base field of the vector space is a finite set?

    Is "a subset U of zero contains a scaled copy of the whole space" the same as "a scaled copy of a subset U of zero is the whole space"?

Thanks and regards!

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    Thanks, @DavidMitra and t.b.!2012-02-24

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