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Let $d((x_1,x_2),(y_1,y_2))=\max \{|x_1-y_1|,|x_2-y_2|\}$.

1. Positivity & identity: $|x_i-y_i|\ge 0\Rightarrow d\ge 0$. If $d=0$ then $|x_1-y_1|=|x_2-y_2|=0\Rightarrow x=y$; conversely $x=y\Rightarrow d=0$.

2. Symmetry: $d(x,y)=\max \{|x_1-y_1|,|x_2-y_2|\}=\max \{|y_1-x_1|,|y_2-x_2|\}=d(y,x)$.

3. Triangle inequality: For any $x,y,z\in {\mathbb{R}}^2$, $|x_i-z_i|\le |x_i-y_i|+|y_i-z_i|(i=1,2).$ Hence, with $A=|x_1-y_1|,B=|x_2-y_2|,C=|y_1-z_1|,D=|y_2-z_2|,$

   \le \max\{A,B\}+\max\{C,D\}=d(x,y)+d(y,z),$$ since for $u_i,v_i\ge 0$, $\max(u_1+v_1,u_2+v_2)\le \max(u_1,u_2)+\max(v_1,v_2)$.

Therefore $d$ satisfies all metric axioms on ${\mathbb{R}}^2$.

Let A = {1/n^2 : n∈ℕ} ⊂ ℝ (Euclidean metric).

Interior(A):

• Each a=1/n^2 is isolated: ∃r>0 with (a−r,a+r)∩A={a}.
• Any open ball around a contains points not in A ⇒ no nonempty open set ⊂ A. ⇒ Int(A)=∅.

Closure(A):

• 0 is a limit point: 1/n^2 → 0 and 0∉A.
• Every other point x≠0 is not a limit point (choose r < ½·min{|1/n^2−x|}). ⇒ Cl(A)=A∪{0}.

Boundary(A):

• ∂A = Cl(A) \ Int(A) = (A∪{0}) \ ∅ = A∪{0}. (Indeed, any ball around a∈A contains a (∈A) and non-A points; any ball around 0 meets A.)

Summary: $\mathrm{Int}(A)=\varnothing ,\overline{A}=A\cup \{0\},\partial A=A\cup \{0\}.$