Equivalence condition for vector addition and scalar multiplication is continuous

Question

My textbook says that vector addition and non-zero scalar multiplication is continuous are equivalent to the following things:

In Vector space $(\mathbb{E},\mathbb{K}),\mathbb{K}=\mathbb{R}\ or\ \mathbb{C}$ and topological space $(\mathbb{E},\tau)$

1. vector addition is continuous at $(0,0)$
2. scalar multiplication is continuous at $(0,0)$
3. $\forall x\in\mathbb{E}$, $\lambda x$ is continuous at $0\in\mathbb{K}$
4. $\forall \lambda\in \mathbb{K}$, $\lambda x$ is continuous at $0\in\mathbb{E}$

I'm not familar with point set topology, so I have difficulty to prove it is a TVS.

Answer 1

I have asked my TA, and he said one should select a particular $\tau$ such that it makes $\forall \alpha\in\mathbb{E},\ T_\alpha:\mathbb{E}\rightarrow\mathbb{E},\ x\mapsto \alpha+x$ and $\forall \lambda\in\mathbb{K}\setminus\{0\}, M_\lambda :\mathbb{E}\rightarrow\mathbb{E},\ x\mapsto\lambda x$ continuous with respect to $\tau$.

And under that condition we can prove the equivalence between i)-iv) and the continuity of $+:\mathbb{E}\times\mathbb{E}\rightarrow\mathbb{E}$ and $\cdot:\mathbb{K}\times\mathbb{E}\rightarrow\mathbb{E}$.