- Thread starter
- Banned
- #1
If $X$ is a metric space, then $X-\emptyset$ is open, then $X$ is closed. Also, $X-\emptyset$ is closed so $X$ is open. So the whole set is both open and closed.Are all metric spaces open sets? How can this be proved?
It is a metric space ie a Topological space. What are the requirements for a topological space?how do you know X without the empty set is closed?
The statement that a set $O$ is open means that if $x\in O$ then some ball $\mathcal{B}(x;\delta)\subseteq O$.I'm learning about metric spaces before I go on to topological spaces. I thought there would be a proof independent of that.
You are mixing up definitions.Well, if the definition doesn't stipulate that X is open, I don't quite see how X is open 'by definition', as Plato claims.