Is there a topological space and meanwhile a linear space such that its vector addition is discontinuous but scalar multiplication is continuous?

Question

The title is the question.

Does there exist a topological space and meanwhile a linear space X such that its vector addition operation is discontinuous but scalar multiplication operation is continuous?

I really find it difficult for me. Help me please. I can't work it out so far.

Answer 1

Yes. Let $V=\mathbb R^2$ and declare $U\subset V$ open iff for all $v\in U$, $a\in\mathbb R$ also $av\in U$.