I am trying to understand topological vector spaces. Is $(\mathbb R,+,0)$ a TVS when equipped with the cocountable topology ?
I mainly have problems to understand continuity of $+$. I think it is possible to make a convergent (in the product topology) net $(x_\lambda,y_\lambda)$ such that $x_\lambda$ and $y_\lambda$ both converge but $x_\lambda+y_\lambda$ not.
Any clarifications are appreciated.
The answer is no, regardless of the field. For example, the set of pairs $(x,y)$ such that $x+y \neq 0$ is not open as it should be if $+$ were a continuous operation. To see this, just note that any cocountable $A \subseteq \mathbb{R}$ contains a pair $\{x,-x\}\subseteq A$.