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let's say that $(X_1,d_1)$ and $(X_2,d_2)$ are two metric spaces. I define $X= X_1 \times X_2$ and a metric $d$, $d((x_1,x_2),(y_1,y_2))=d_1(x_1,y_1)+d_2(x_2,y_2)$.

I want to prove that for an open part $A_1 \subset X_1$ and an open part $A_2 \subset X_2$, $A_1 \times A_2 $ is an open part of $X_1 \times X_2$ for the metric $d$.

I find this really difficult to prove because I don't know how to use $d$.

I thought because $A_1$ open in $X_1$ for $d_1$ there exist an $\delta_1$ so that an open $B_1(x,\delta_1)=\{y \in X_1 | d_1(x,y)<\delta_1 \} \subset A_1$ for $x\in A_1$ and there exist an $\delta_2$ so that an open $B_2(x,\delta_2)=\{y \in X_1 | d_1(x,y)<\delta_2 \} \subset A_2$ for $x\in A_2$.

Now I don't know how to use the cartesian product?

questmath
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1 Answers1

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Take $(a_1,a_2) \in A_1\times A_2$. Let $B_1(a_1,\delta_1) \subset A_1$ and $B_2(a_2,\delta_2) \subset A_2$. Let $\delta =\min \{\delta_1,\delta_2\}$. Verify that $B_{d} ((a_1,a_2), \delta) \subset A_1 \times A_2$.

Henno Brandsma
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Kavi Rama Murthy
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