The diagram you show is demonstrating that any inner product space is a normed vector space, any normed vector space is in turn a metric space, and any metric space is in its turn a topological space.
A topological space is not a vector space because... well, it's just not. It doesn't satisfy the things it's required to satisfy in order for it to be a vector space. I give you for example the so called Sierpinski Space. This is clearly not a vector space.
A vector space is in turn not a topological space unless you define a topology on it. The comment from Jacky is explaining that given any vector space, you could for example give it the discrete topology, thus giving you a vector space which is also a topological space.
In general, however, a vector space isn't a topological space.