3

In Metric is continuous function, I found the following answer

Define $f:X\times X\to \mathbb R$ by $f(x,y)=d(x,y)$.

Let $(x_n,y_n)$ be a sequence in $X\times X$ such that $(x_n,y_n)\to (x,y)$

Then $(x_n)\to x,(y_n)\to y\implies d(x_n,x)\to 0 ,d(y_n,y)\to 0 $ as $n\to \infty\implies d(x_n,y_n)\leq d(x_n,x)+d(x,y)+d(y_n,y)\to d(x,y)$

$\implies d(x_n,y_n)\to d(x,y) $ as $n\to \infty \implies f(x_n,y_n)\to f(x,y)$

The above proof depends on the fact that the metric on $X\times X$ is so chosen that $(x_n,y_n)\to (x,y) \implies x_n \to x$ and $y_n\to y$. I have a query whether there is a metric on $X\times X$ so that the metric function $d:X\times X \to \mathbb{R}$ is not continuous.

LoveMath
  • 109
  • Please rephrase your last sentence with the most formailty. In particular, continuous wrt what? – SBF Apr 20 '23 at 11:13
  • 1
    @Ilya it seems pretty clear (to me at least) that they are asking about the existence of a metric $m$ on $X \times X$ such that $d$ is not continuous from $(X \times X, m)$ to $(\mathbb{R}, |\cdot|)$? – Bruno B Apr 20 '23 at 11:18
  • Linking an identical question just for duplicate flagging sake. Though I do not understand why that question is itself flagged for being a duplicate when the post it links to does not answer the question... https://math.stackexchange.com/q/4497731/1104384 – Bruno B Apr 20 '23 at 11:34

1 Answers1

4

An example where the metric on $X\times X$ is such that $(x_n,y_n)\to (x,y) \nRightarrow x_n \to x$ and $y_n\to y$:

Hint: Think of a (discontinuous) bijection $f$ of the real line with $x_n \to x, f(x_n) \to y=f(z)\neq f(x)$. Let $d((x,y),(u,v))=|f(x)-f(u)|+|f(y)-f(v)|$. Then $(x_n,0) \to (z,0)$ but $x_n$ does not tend to $z$.

Jean-Claude Arbaut
  • 23,277
geetha290krm
  • 36,632
  • Thanks. This means that the distance function is not continuous if $X\times X$ is equipped with this metric. Is it so? Is the topology generated by this metric different from the product topology on $X\times X$? – LoveMath Apr 21 '23 at 06:59
  • @LoveMath Yes, it is different from the product topology. – geetha290krm Apr 21 '23 at 07:23