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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.

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  • $\begingroup$ For the first question, try the paris metric. In particular, let $\mathbb{R}^n$ be viewed as a real vector space, and let $d_E$ denote the usual Euclidean metric on $\mathbb{R}^n$. Then we can define a new distance function $d$ on $\mathbb{R}^n$ by asserting that $d(x,y)=d_E(x,y)$ if $x$ and $y$ are parallel, otherwise $d(x,y) = d_E(x,0)+d_E(0,y).$ Try googling "Paris Metric Space" for more information. $\endgroup$ – goblin Jan 10 '14 at 10:11
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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.

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  • $\begingroup$ Now that I think about it, it is possible that by "scalar" multiplication you actually mean some sort of an inner product. In my native tongue inner products are literally called "scalar", which creates confusion. If that's the case, my answer is irrelevant. $\endgroup$ – Dan Shved Jan 10 '14 at 10:18
  • $\begingroup$ Wa, what you have said is very helpful to me. And what you have mean is just what I have just questioned. But could you solve this problem: "Is there a metric space and meanwhile a group G such that its product operation is continuous but inverse operation is dis-continuous?" or the second question" 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?" These problems are so difficult for me. $\endgroup$ – David Chan Jan 24 '14 at 2:58

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