For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an $\epsilon > 0$ such that $x + \alpha v \in O$ for every $\alpha$ with $| \alpha | < \epsilon$. This gives a topology indeed (right?). I think that in finite-dimensional vector spaces, this gives the standard topology (right?).
What about infinite-dimensional vector spaces? Is there a name for this topology? To me, this looks like a very generic way to define a topology on a vector space, but I've never seen this definition of a topology. Is there any connection to locally convex topologies? (I'm not very familiar with this concept) Is this topology important in any application?
This is not an answer. Rather, let me explain another canonical way of endowing vector spaces with a topology.
Let $V$ be a vector space over the real numbers $\mathbf{R}$. (You can replace $\mathbf{R}$ by any topological field in everything that follows, although for the properties at the end you'll need your topological field to be complete of characteristic zero and Hausdorff.)
Suppose that $V$ is finite-dimensional. Any isomorphism $V\cong \mathbf{R}^n$ endows $V$ with a topology compatible with the vector space structure, where we endow $\mathbf{R}^n$ with the product topology. (Such an isomorphism boils down to the choice of a basis for $V$.) It is easy to see that this topology is independent of the chosen isomorphism.
Now, in general, if $V$ is not finite-dimensional, you can endow $V$ with the "inductive limit topology". This topology is characterized by the following property:
A map $f:V\to A$ is continuous if and only if its restriction $f|_{W}:W\to A$ to every finite-dimensional subspace $W$ of $V$ is continuous.
It is a direct consequence of the definitions that the inclusion $W\to V$ of a finite-dimensional subspace $W$ of $V$ is continuous. More generally, any $\mathbf{R}$-linear map $V_1\to V_2$ of vector spaces endowed with the inductive limit topology is continuous.
Properties:
Any vector space of $V$ is closed.
The topology is Hausdorff.
Let $K\subset V$ be compact. Then $K$ is contained in a finite-dimensional subspace of $V$.
The first two properties are not so hard to prove. In fact, to prove that $V$ is Hausdorff proceed as follows. Let $x$ and $y$ be in $V$. Consider the subspace $W$ generated by $x$ and $y$. Choosing a basis for $V$ gives you a projection $p:V\to W$. This projection is continuous. Let $U_x$ be an open of $W$ containing $x$ and $U_y$ an open of $W$ containing $y$ such that $U_x\cap U_y$ is empty. (Here you use that the base field is Hausdorff.) Then $p^{-1}(U_x)$ and $p^{-1}(U_y)$ separate $x$ and $y$ in $V$.
The third one is a bit trickier. You'll need to use that the base field is complete and of characteristic zero.