Examples of metric spaces in which every non-empty open set is uncountable

Question

Is every non-empty open set of a complete metric space uncountable ? If not can anyone please provide some examples of metric spaces (other than $\mathbb R$ with usual metric) in which every non-empty open set is uncountable ?

Answer 1

Every open set in $\mathbb R^n$ is uncountable: if $U$ is open in $\mathbb R^n$, then for every $x$ in $U$ there is a ball $B(x,r) \subset U ; r>0$ (since the open balls are a basis for the metric topology of $\mathbb R^n$). You can use, e.g., stereographic projection to show that these balls contain uncountably-many points. On the other extreme, in any discrete metric space , singletons are open ( as is any collection of points, which is also closed.) Maybe trivially, same thing applies for $\mathbb C^n$, since $\mathbb C^n$ is homeomorphic to $\mathbb R^{2n}$.

Answer 2

To answer your first question

Is every non-empty open set of a complete metric space uncountable ?

This is false, there are in fact countable complete metric spaces. For instance the discrete metric on any countable set forms a complete metric space. Also something like $\{0,1,1/2,1/3,1/4,...\}$ with the Euclidean metric is a complete metric space. We can also find uncountable complete metric spaces that have countable non-empty sets by taking again the discrete metric on an uncountable set.