1. ## metric continuous function

Suppose $\displaystyle (\mathbb{X},d)$ is a metric space and the corresponding product topology is defined on $\displaystyle \mathbb{X}\times\mathbb{X}$. Prove that the metric d is continuous.

2. Originally Posted by raimis
Suppose $\displaystyle (\mathbb{X},d)$ is a metric space and the corresponding product topology is defined on $\displaystyle \mathbb{X}\times\mathbb{X}$. Prove that the metric d is continuous.
You have to show that if $\displaystyle (x,y)$ is close to $\displaystyle (x_0,y_0)$ in $\displaystyle \mathbb{X}\times\mathbb{X}$, then $\displaystyle d(x,y)$ is close to $\displaystyle d(x_0,y_0)$. More precisely, you have to show that, given $\displaystyle (x_0,y_0)\in\mathbb{X}\times\mathbb{X}$ and $\displaystyle \varepsilon>0$, there exists $\displaystyle \delta>0$ such that if $\displaystyle d(x,x_0)<\delta$ and $\displaystyle d(y,y_0)<\delta$ then $\displaystyle |d(x,y) - d(x_0,y_0)|<\varepsilon$. Use the triangle inequality.

3. ## product topology

And how do we make use of the product topology here?

4. Originally Posted by raimis
And how do we make use of the product topology here?
The condition $\displaystyle d(x,x_0)<\delta$ and $\displaystyle d(y,y_0)<\delta$ expresses the fact that $\displaystyle (x,y)$ is close to $\displaystyle (x_0,y_0)$ in the product topology.

5. Originally Posted by raimis
Suppose $\displaystyle (\mathbb{X},d)$ is a metric space and the corresponding product topology is defined on $\displaystyle \mathbb{X}\times\mathbb{X}$. Prove that the metric d is continuous.
I'm sorry. What is this question actually asking. I see that Opalg has already answered it, but I am just curious. Is it that if $\displaystyle (X,d)$ is a metric space then $\displaystyle d:X^2\mapsto\mathbb{R}$ is continuous?

6. Originally Posted by Drexel28
Is it that if $\displaystyle (X,d)$ is a metric space then $\displaystyle d:X^2\mapsto\mathbb{R}$ is continuous?
Yes, the question is as you say. But the problem involves product topology and I am wondering whether it should necessarily be defined on $\displaystyle \mathbb{X}\times\mathbb{X}$ to prove the statement.