Question about metric function

Question

Suppose that $(X,d)$ is a metric space. Prove that $d:X\times X\to \mathbb{R}$ is a continuous.

Remark: I know that there a lot of similar topics such as this question. Please do not duplicate because the question which I am going to ask I did not meet in other topics.

Let $(x_0,y_0)$ some point from $X\times X$. And I am going to prove that the function $(x,y)\mapsto d(x,y)$ is continuous at $(x_0,y_0)$. We need to show that for any $\varepsilon>0$ $\exists \delta=\delta(\varepsilon)>0$ such that for any $(x,y)\in X\times X$ which is close to $(x_0,y_0)$ by $\delta$ we have distance between $d(x,y)$ and $d(x_0,y_0)$ is less than $\varepsilon$.

But I have the following question:

1) What is the distance between $(x,y)$ and $(x_0,y_0)$ which are points of $X\times X$?

2) What is the distance between $d(x,y)$ and $d(x_0,y_0)$ which are real numbers?

I would be very grateful for explanation!

Answer 1

There are many different distances that one can put on $X\times X$ and $\mathbb{R}$.

Typically, the distance on $\mathbb{R}$ is $d(\alpha,\beta)=|\alpha-\beta|$. If $(X,d)$ is a metric space, one possible distance on $X\times X$ is $d((x,y),(x_0,y_0))=\max(d(x,x_0),d(y,y_0))$.

Answer 2

You are quite right to question what metrics or topologies you should be using when you judge whether $d : X \times X \to \Bbb{R}$ is continuous. The product topology on $X \times X$ (where $X$ is given the metric topology induced by $d$) and the standard topology on $\Bbb{R}$ are the ones that make sense.