Example of a vector space that is not a topological space?

Question

I saw this on Wikipedia about the hierarchy of spaces:

Later in the article, it states that the fundamental building blocks of mathematical spaces are vector spaces and topological spaces. However, it's not clear to me how topological spaces and vector spaces are related, if at all.

Can someone provide an example of a vector space that is not a topological space and vice versa?

Answer 1

Vector spaces and topological spaces are fundamentally different concepts; one is a set with and addition and scalar multiplication, the other a set together with a set of subsets (the open sets) with certain constraints.

Answer 2

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.

Answer 3

Your question is a little subtler than you think. This hierarchy is about "spaces" that have a geometric structure. The inner product spaces that are an example of all the containing kinds of spaces have geometric structure that's compatible with the vector space axioms. At the top level that means that vector addition and scalar multiplication are continuous functions in the topology defined by the inner product in the spaces at the bottom of the hierarchy.

To address the exact question you asked, let $V$ be any vector space. Then thinking of $V$ as just a set (ignoring the vector space structure), you can equip it with any topology you like.

There are topological spaces that can't be made into vector spaces. Consider a $6$ element set, with the discrete topology. That can't be given a vector space structure - not even over a finite field.