Question

Let (x,d) be metric space with the discrete metric .prove that every subset of X is open

Answer 1

Solution:

Proof:

Let (X, d) be a discrete metric space. Thus singletons are open sets as $\{x\} = B(x,\epsilon )$ where $\epsilon<1$ . Any subset A can be written as union of singletons. As any union of open sets is open, any subset in X is open.

Hence, proved.