Topology on $\mathbb{K}\times E$

Question

This is probably a really basic question, but here it goes. Let $E$ be a vector space over $\mathbb{K}$ and let $\tau$ be a topology on $E$. Then $E$ is called a topological vector space if the sum and scalar multiplications are continuous, i.e. if the maps $s: E\times E \to E$ and $m: \mathbb{K}\times E \to E$ defined by $s(x,y) := x+y$ and $m(\lambda, x) := \lambda x$ are continuous.

The topology on $E\times E$ is obviously the product topology but what is the topology on $\mathbb{K}\times E$?

Answer 1

Typically $\mathbb{K}$ would be $\mathbb{R}$ or $\mathbb{C}$, which have natural topologies already. Then the product topology is what is meant for $\mathbb{K} \times E$. If $\mathbb{K}$ were something else, it would need a prescribed topology for this to make sense.

Answer 2

We give $\mathbb{K}=\mathbb{R},\mathbb{C}$ the typical topology (induced for example by $|\cdot |$) and let $\mathbb{K}\times E$ have the product topology.