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Let $A = \{a+bi \in \mathbb{C} : a > 4\}$ and $B = \{ a+bi \in \mathbb{C} : a \geq 4\}$ and let $d(z,w)$ be the metric on $\mathbb{C}$ defined as,

$$d(z,w) = \begin{cases} 0 &, \mathrm{if}\, z =w\\ |z| +|w| &, \mathrm{if}\, z\neq w \end{cases}$$

My questions are:

  1. Are $A$ and $B$ open in $(\mathbb{C},d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{C}$?
  2. Are $A$ and $B$ open in $(\mathbb{C},d)$, where $d$ is defined as above?

My solution:

  1. From my understanding, $A$ is open, as it does not contain its boundary point $\{4\}$ and its complement is closed. $B$ is not open as it contains its boundary point $\{4\}$ and its complement is open.
  2. I am stuck here however as the answer says both are open? But how do I prove this?
David Gao
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Saurav Sathnarayan
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  • For the usual metric, $A$ is indeed open ($\iff$ its complement is closed) but your argument is not sufficient ($4$ is not the only boundary point). And $B$ is closed ($\iff$ its complement is open) but you only proved that $B$ is not open. – Anne Bauval Mar 12 '24 at 23:16
  • Where are you stuck in your attempt for the second question? The natural way is to start from the definitions: when is a subset of a metric space open? And if $r<|z|$, what is $B_r(z)$? – Anne Bauval Mar 12 '24 at 23:29
  • @AnneBauval, I changed the metric now. The question was vague, so proving the set was closed was not needed, i only needed to prove if it was open. – Saurav Sathnarayan Mar 13 '24 at 11:04
  • For the second part, i cant seem to intuitively understand this metric. I understand under the usual metric why the set A is open and B is not open. But under this modified metric, why are they both open. – Saurav Sathnarayan Mar 13 '24 at 11:06
  • It is the "French railway metric" (the shortest way between two towns goes through Paris), but you don't need this "intuitive understanding". Just apply the standard recipe recalled in my second comment. A look at the linked post may also help. – Anne Bauval Mar 13 '24 at 11:08
  • Are you familiar with the open ball definition of an open set? A set is open if all its points are Interior Points. A set is closed if it contains all of its Limit Points, in particular those Limit points which are it's Boundary points. Metrics determine which are limit points and which are interior points. For example, the "Parisian" metric will not allow non-zero limit points. No two points can be closer together than the largest modulus of one of those points. So there are no limit points. – TurlocTheRed Mar 20 '24 at 04:24

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