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!

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 $$d((x,y),(x_0,y_0))=\max(d(x,x_0),d(y,y_0))$$.

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.

• The base of metric topology on $X$ is the collection of all balls $\{B_d(x,\varepsilon): x\in X, \varepsilon>0\}$. So the base of product topology on $X\times X$ is the product of open sets in $X$, right? So what is the distance between $(x,y)$ and $(x_0,y_0)$. Could you clarify your answer a bit, please?
– ZFR
Oct 15 '18 at 23:22
• There are many different metrics that result in the product topology on $X \times X$ (Scientifica's answer tells you about one of them). If you haven't been taught about the product topology or been told which metric to use, then the question you've been asked is a bit unfair on you. On the other hand, if you do know about the product topology (and your comment suggests you do), then you don't need to work with a metric on $X \times X$, instead you can just prove continuity with respect to the product topology. Oct 15 '18 at 23:30
• Yes I know some facts about product topology. What does mean your last sentence? Do you mean to prove that $d^{-1}(a,b)$ is open in $X\times X$? Sorry if the question sounds dumb because I began to study topology recently.
– ZFR
Oct 15 '18 at 23:44
• Your comment isn't dumb and you have identified exactly what you need to do to show that $d$ is continuous. Oct 15 '18 at 23:51