From Wikipedia:
"Let $K$ be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications $K$ will be either the field of complex numbers or the field of real numbers with the familiar topologies."
I have two questions:
Can you give me an example of a topology on $K$ such that addition is not continuous? I assume that addition is continuous means that $f(x,y) = x + y$ is continuous in both $f(x, \cdot)$ and $f(\cdot, y)$.
The other question is: how is division continuous? I assume again that division by $y$ is a function $f_y(x) = \frac{x}{y}$. How is $f_0$ continuous in the standard topology?
Sure, you can put a terrible topology on $K$ and then addition won't be continuous. For example, pick the cofinite topology on $\mathbb{R}$ if you like. The preimage of the origin, which is a closed point, under addition $\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ will be the antidiagonal set $\{(x, -x)\} \subset \mathbb{R}^2$, which is infinite and therefore not closed.