Topology: An Inquiry-Based Approach

If we have two metric spaces $(X,d_1)$ and $(X_2,d_2)\text{,}$ we might wonder if we can make the set $X_1\times X_2$ into a metric space. A natural approach might be to define a function $d:(X_1\times X_2)\times (X_1\times X_2)\to \mathbb{R}$ by

for $(x,y)$ and $(u,v)$ in $X_1\times X_2\text{.}$ However, this $d$ does not define a metric. For example, if $x\in X_1$ and $y\ne v$ in $X_2\text{,}$ then $d((x,y),(x,v))=0$ even though $(x,y)\ne (x,v)\text{.}$ To make a metric, we can take a clue from the Euclidean metric on $\mathbb{R}\times \mathbb{R}\text{.}$ On $\mathbb{R}\text{,}$ the metric has the form $d_1(x,y)=|x-y|\text{,}$ while on ${\mathbb{R}}^2$ the metric is

So on $(X,d_1)$ and $(X_2,d_2)$ we could consider defining $d:(X_1\times X_2)\times (X_1\times X_2)\to \mathbb{R}$ by

Let $x$ and $y$ be defined as in Activity 11.3, and let $z=(z_1,z_2)$ be in $X_1\times X_2\text{.}$ Then

Since all terms are non-negative we conclude that