Disclaimer: I'm aware this is a duplicate of Prove that open half planes are open sets.
However, I didn't find an adequate answer on that page, and it is a year old.
The top answer for the question by graydad is too restrictive in my opinion because it relies on the metric function being the Euclidean distance, but I'm pretty sure a more general proof should be possible without relying on the exact computation of the metric function. (Couldn't we arrive at the same conclusion that the half-plane is open even by using any arbitrary metric like the French Metro Metric, or the Taxicab norm, or any other conceivable metric on $\mathbb{R}^2$?)
The problem is to prove that the half-plane $H_a = \{(x,y) \in \mathbb{R}^2 : x>a\}$ for any $a \in \mathbb{R}$ is an open set using any arbitrary metric function on $\mathbb{R}^2.$
I am assuming definitions of open and metric function as given in chapter 2 of Rudin:
Rudin, Walter. Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics). 3rd ed. McGraw-Hill, 1976. ISBN: 9780070542358.
In particular, the definition of open set is:
A set $E$ is open if every point in $E$ is an interior point.
A point $p$ is an interior point of $E$ if there is a neighborhood $N$ of $p$ such that $N \subset E$.
A neighborhood of a point $p$ is a set $N_r(p)$ consisting of all points $q$ such that $d(p,q)<r$. The number $r$ is called the radius of $N_r(p)$.
and the definition of metric is:
A function that associates with any two points $p$ and $q$ a real number $d(p,q)$ such that:
$d(p,q)>0$ if $p \ne q$ else $d(p,q)=0$
$d(p,q)=d(q,p)$
$d(p,q)≤d(p,r)+d(r,q)$ for any $r \in X$.
(Also any proof should use only the most basic principles… Specifically material from the first two chapters of Rudin.
That's why the other answer by Aloizio Macedo in the original post was not helpful for me.)
Edit: I removed my attempted proof using relative topology because it was too long and not helpful.
In the comments Yujie Zha suggested I do a direct proof using just the definition of open set. However, I also tried this with no success. There is probably an easy way to write a direct proof for this but I don't see it…
Here's roughly my train of thought:
Let $p=(x,y)$ be a point in $H_a$. Let $N_r(p)$ be a neighborhood of $p$ with radius $r=x−a$. (Now I would like to say $N_r(p) \subset H_a$, but how? Is it by the triangle inequality somehow?) Suppose there is a point $q=(x′,y′) \in N_r(p)$ that is not in $H_a$. Then $x′<a$. But… how do I arrive at the necessary contradiction?
Best Answer
Your comments do, perhaps, clarify what you are asking somewhat. All of the metrics you mention specifically in your comment (the Euclidean metric; the French metro metric; the taxicab metric) determine the exact same collection of open subsets of $\mathbb{R}^2$ (this is a good exercise). So, open-ness of a half plane with respect to any one of those three metrics is equivalent to open-ness of that half plane with respect to any other of those three metrics. This collection of open sets is known as the "standard topology on $\mathbb{R}^n$", as you will learn if you read a book or take a course on topology. So, to prove open-ness with respect to any one of these metrics, feel free to pick one of those three metrics, and prove open-ness just for that one.
Now, you also ask "how to show that a half-plane is open with a general metric on $\mathbb{R}^2$".
Well, that's false: a half-plane is not open with an arbitrary metric on $\mathbb{R}^2$.
Here's a counterexample. Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be the bijection which equals the identity except that $f(0,1)=(0,-1)$ and $f(0,-1)=(0,1)$. Define a metric by $$d(p,q) = |f(p)-f(q)| $$ To put this another way, take your points $p,q$, find their images under $f$, and apply the Euclidean metric to their images. A subset $A \subset \mathbb{R}^2$ is open with this metric if and only if $f(A)$ is open with the Eucliean metric. Hence, the half plane $H$ defined by $y>0$ is not open, because its image under $f$ equals $(H-\{(0,1)\}) \cup \{(0,-1)\}$ which is not open in the Euclidean metric.