# A subset that can be scaled to be the whole space, or that can contain a scaled version of the whole space

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"?

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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well, a scaled copy of the whole space is the whole space! –  t.b. Feb 24 '12 at 16:28
@t.b.: Really, in both TVS and vector space? Then Mariano introduced some unnecessary complication to that quote. –  Tim Feb 24 '12 at 16:30
@DavidMitra: How is the first equality true? –  Tim Feb 24 '12 at 16:32
Sorry, I should have said: $\alpha V=V$, for $\alpha\ne0$. One inclusion is obvious. For the other inclusion, note if $v\in V$, then $v=\alpha(\alpha^{-1}v)\in\alpha V$. –  David Mitra Feb 24 '12 at 16:40
Thanks, @DavidMitra and t.b.! –  Tim Feb 24 '12 at 18:40