Definition of Absorbing Set in Topology Vector Space

[AxiomOfChoice]

Is this a legitimate definition for an "absorbing set" in a topological vector space?

A set A\subset X is absorbing if X = \bigcup_{n\in \mathbb N} nA.

This is the definition the way it was presented to us in my functional analysis class, but I'm looking at other sources, and it seems everyone talks about absorbing being something that can be arbitrarily expanded (by *any* scalar) to include the whole space. It seems that expansion just by natural numbers is too restrictive.

 

[AxiomOfChoice]

My professor also claims that EVERY neighborhood of zero in a topological vector space is absorbing. This wasn't proved, and I can't think of why it would be the case.

 

[JSuarez]

Because if X = \bigcup_{n\in \mathbb N} nA, then any element of X eventually belongs (is absorbed) by a scaled version nA of A. A little thinking also shows that any open neighborhood is absorbing.