From Wikipedia:
Given a topological vector space $(X,τ)$ over a field $F$, $S$ is called bounded if for every neighborhood $N$ of the zero vector there exists a scalar $α$ so that $ S \subseteq \alpha N $ with $ \alpha N := \{ \alpha x \mid x \in N\}. $
I was wondering if the concept is still the same when "for every neighborhood $N$ of the zero vector" is replaced by "there exists a neighborhood $N$ of the zero vector"? Is it true that every neighborhood of the zero vector can be scaled to contain any other neighborhood of the zero vector?
Thanks and regards!