Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous? Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's difficult to give some examples s.t. some conditions.So I put the things listed down.
1.Is there a metric space and meanwhile a linear space X such that its vector addition operation is discontinuous but scalar multiplication operation is continuous? There must exist. However, this question is difficult for me to solve it. I have tried my best to think about it, but gained nothing. 
2.Of course, I also don't know that if 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?
3.Does there exist a metric space and meanwhile a group X such that its multiplication operation is continuous but inverse operation is discontinuous?
4.For a topological group, I'd like to know whether there exist a topological group G which is a Hausdorff space but does not satisfies the first countable axiom or there exist a topological group G which is not a Hausdorff space and does not satisfies the first countable axiom.
5.For a topological vector space(tvs), I'd like to know whether there exist a topological vector space V which is a Hausdorff space but does not satisfies the first countable axiom or there exist a topological vector space V which is not a Hausdorff space and does not satisfies the first countable axiom.
Maybe I need six different examples to prove some important conclusions.
help me please.That's really difficult for me.
 A: These are many questions, I will answer only the first one (it is also in the question title).
Let $X = \mathbb{R}^2$. It is a vector space over $\mathbb{R}$. Denote by $|\cdot|$ the standard norm $|(x_1, x_2)| = \sqrt{x_1^2 + x_2^2}$. Now endow $X$ with a metric structure like this: for every $u, v \in \mathbb{R}^2$ set
$$
d(u, v) = \begin{cases}|u| + |v| & \text{if $u$ and $v$ are linearly independent;}\\ |u-v| &\text{otherwise.}\end{cases}
$$
You can check that this is indeed a metric. To understand it intuitively, think of $\mathbb{R}^2$ as a "star" with center $(0, 0)$ and with infinitely many rays going in each direction. You can travel along these rays, and $d(u, v)$ is the length of the shortest path from point $u$ to point $v$.
Now, I claim that under this metric the addition is not continuous, but multiplication is (in the sense that map $(\lambda , u) \to \lambda \cdot u$ is a continuous map $\mathbb{R}\times X \to X$). I'm not proving these claims, but I'd like to point out that they are intuitively clear. The "star" that I'm talking about is preserved under multiplications, but isn't preserved under addition.
