4
$\begingroup$

Is it possible that in a metric space $(X, d)$ with more than one point, the only open sets are $X$ and $\emptyset$?

I don't think this is possible in $\mathbb{R}$, but are there any possible metric spaces where that would be true?

1 Answers 1

14

One of the axioms is that for $x, y \in X$ we have $d(x, y) = 0$ if and only if $x = y$. So if you have two distinct points, you should be able to find an open ball around one of them that does not contain the other.

  • 0
    @DylanMoreland Got it! And not too hard to prove too! Thanks for the help! I'm just starting to self study topology so am rough on the basics.2011-11-30