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Define $d:\mathbb{R}^2\times \mathbb{R}^2\to \mathbb{R}$ by

$d(x,y)=||x-y||$ if $x=ty,t\in \mathbb{R}$ 

         otherwise ||x||+||y||

(a) Describe the basis of open balls. (Be careful as the radius of a ball grows.)

So if the vectors are parallel, we have the euclidean norm, but if they are not we have the sum of each vectors norm.

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    The usual distance is used if the two points $x$ ,$y$ lie on one line with $0$ (radial). Otherwise the distance is the sum of norms (we go via $0$).SO if $x = (1,1)$ e.g. the only points with distance $ r < \sqrt{2} = ||x||$ are the points within that distance that lie on the line $y = (t,t), t \in \mathbb{R}$. – 2017-01-26
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    What you wrote is most likely what I would have answered? Is it really that kind of answer they want tho? I'd expect something like to describe what the balls look like(i.e a circle, a square or something else) – 2017-01-26
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    Small balls (enough for a base) are radial open lines. Only balls around the origin are the usual balls – 2017-01-26

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