# Partitioning a compact open set into balls in an ultrametric space

Consider a $p$-adic field $K$ with the standard topology inherited from the usual $p$-adic norm $\mid \cdot \mid$. Consider the ultrametric space $X=K^n$ with the topology inherited from the norm $\| \cdot \|$ defined as $\|x\|=\max_{i=1}^n (|x_1|,\dots,|x_n|)$ with $x=(x_1,\dots,x_n) \in K^n$. Now we have two questions:

1. Is the topology on $X$ the same as the product topology of the $K$'s?

2. Can we partition every open compact set in $X$ into a finite number of balls (defined using $\|\cdot \|$)?

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1. is true because the sets $U_{\epsilon} = \{(x_1,\dots,x_n),\ |x_i| \leq \epsilon\}$ for a basis of neighborhood of $0$ in $X$ for your norm by definition, but also a basis of neighborhood of $X$ for the product topology since any neighborhood of $0$ in the product topology contains a product of neighborhoods of $0$ which each contains a ball $|x_i| \leq \epsilon_i$ and it suffices to take $\epsilon=\min(\epsilon_i)$.
Thank you Joel. I think your answer will help me to settle down another question on $p$-adic semilagebraic isomorphisms. –  user16974 Oct 27 '11 at 15:13