How to find a 'nontrivial' open cover of an arbitrary metric space?

Question

Give a 'nontrivial' open cover of an arbitrary metric space. This question is from topology metric spaces by S. Kumaresan (page 82 excercise 4.1.7). I don't have any idea what i have to do.

Answer 1

An open cover of a metric space $X$ is a collection of open subsets $(U_i)_i$ such that $X = \bigcup_i U_i$. As $X$ is an open set, the cover $\{X\}$ of $X$ is welldefined. Any cover containing $X$ as one of the open subsets is considered to be trivial. Hence you are asked to find a cover consisting purely of proper open subsets. Can you give such a cover?

Answer 2

Consider the family of all the open subsets of $X$, $\tau = \{ U \subseteq X :\, U \textrm{ is open in } X\}$. Surely $\tau$ is an open cover of $X$, but also, $\{X\}$ is an open cover of $X$. So, this two open covers can be considered trivial, right? (Yeah, that exercise is very weird).

Just an idea: can you prove that $$\mathcal O_r := \{ B(x,r) :\, x \in X \} \quad \textrm{for some $r \in \mathbb R^+$}$$ is an open cover of $X$?