metric continuous function

• Feb 15th 2010, 08:07 AM
raimis
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.
• Feb 15th 2010, 11:32 AM
Opalg
Quote:

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.
• Feb 15th 2010, 11:48 AM
raimis
product topology
And how do we make use of the product topology here?
• Feb 15th 2010, 01:04 PM
Opalg
Quote:

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.
• Feb 15th 2010, 07:08 PM
Drexel28
Quote:

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?
• Feb 15th 2010, 10:47 PM
raimis
Quote:

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.