0

In the discrete metric on $\Bbb{R}$ , find the interior, boundary, and closure of $(1,2]$.

I know that in the discrete metric, all singletons are open and closed sets, and all subsets are both open and closed.

I have that:

Interior: {2}

Boundary: {1}

Closure: {1,2}

Is this correct?

kt046172
  • 545
  • Why are closed sets just points? if $A^c$ is open then $A$ is closed, by what you stated above. Also what you wrote about $(1,2]$ doesn't seem right just going by what you wrote before. Perhaps write what you think the definition of interior boundary and closure. – Keen-ameteur Oct 13 '19 at 19:41

1 Answers1

2

In the discrete metric, all subsets are open (as unions of singletons) so all subsets are closed too (all complements are open).

The interior of $A$ is the largest open subset of $A$, so that's $A$ itself for any $A$.

Likewise the closure is the smallest closed superset of $A$, again $A$ itself, always.

As $\partial A=\overline{A}\setminus A$, $\partial A=A \setminus A =\emptyset$ for all $A$.

Henno Brandsma
  • 242,131