Prove that d is a continuous real valued function on $M \times M$

Question

Let $(M,d)$ be a metric space. Let $M \times M$ be the product space, where $d$ is defined (earlier in the book as ) $d((x_1,x_2),(y_1,y_2))=d_1(x_1,y_1)+d_2(x_2,y_2)$ where $d_1$ and $d_2$ are metrics on $M_1$ and $M_2$, respectively.

I must prove that $d$ is a continuous real valued function.

I am very confused about the notation, an I supposed to show that $M \times M$ is the set or $M_1 \times M_2$ is the set in supposed to use? I guess it doesn't really matter but it would really help me.

Can someone help me out on starting the proof?

I know I must choose my delta to be $\varepsilon/2$ since I have two points, but my question is how will some of the notation look like?

Answer 1

Let $d$ denote the metric on $M$ .

Let $\rho$ denote the metric on $M\times M$ wich is defined as $$\rho\left(\left(x_{0},y_{0}\right),\left(x_{1},y_{1}\right)\right)=d\left(x_{0},x_{1}\right)+d\left(y_{0},y_{1}\right)$$

Then it can be shown that $$\left|d\left(x_{0},y_{0}\right)-d\left(x_{1},y_{1}\right)\right|\leq\rho\left(\left(x_{0},y_{0}\right),\left(x_{1},y_{1}\right)\right)\tag1$$

This on base of $$d\left(x_{1},y_{1}\right)\leq d\left(x_{1},x_{0}\right)+d\left(x_{0},y_{0}\right)+d\left(y_{0},y_{1}\right)=\rho\left(\left(x_{0},y_{0}\right),\left(x_{1},y_{1}\right)\right)+d\left(x_{0},y_{0}\right)$$ and: $$d\left(x_{0},y_{0}\right)\leq d\left(x_{0},x_{1}\right)+d\left(x_{1},y_{1}\right)+d\left(y_{1},y_{0}\right)=\rho\left(\left(x_{0},y_{0}\right),\left(x_{1},y_{1}\right)\right)+d\left(x_{1},y_{1}\right)$$

To be shown is that function $d:M\times M\to\mathbb{R}$ is continuous when $\mathbb{R}$ is equipped with its usual topology.

The topology on $\mathbb{R}$ has the intervals $\left(-\infty,a\right)$ and $(a,\infty)$ as subbasis, so it is enough to prove that $\left\{ \left(x,y\right)\in M\times M\mid d\left(x,y\right)<a\right\}$ and $\left\{ \left(x,y\right)\in M\times M\mid d\left(x,y\right)>a\right\}$ are open sets in $M\times M$ for every $a\in\mathbb R$.

That means that for any pair $\left(x_{0},y_{0}\right)$ with $d\left(x_{0},y_{0}\right)<a$ some $\epsilon>0$ must be found such that $\rho\left(\left(x_{0},y_{0}\right),\left(x_1,y_1\right)\right)<\epsilon\Rightarrow d\left(x_1,y_1\right)<a$.

And likewise that for any pair $\left(x_{0},y_{0}\right)$ with $d\left(x_{0},y_{0}\right)>a$ some $\epsilon>0$ must be found such that $\rho\left(\left(x_{0},y_{0}\right),\left(x_1,y_1\right)\right)<\epsilon\Rightarrow d\left(x_1,y_1\right)>a$.

This can be done by use of (1).