Talk:Neighbourhood of Point Contains Point of Subset iff Distance is Zero

Is it true that this theorem only applies to Limit Points? Because if $x$ is an isolated point of $A$, then it can't be a limit point of $A$. But $d \left({x, A}\right) = 0$ because by definition of an isolated point $x \in A$. So $\displaystyle \inf_{y \mathop \in A} d \left({x, y}\right) = d \left({x, x}\right) = 0$.

I think this theorem should apply to the closure of $A$. --HumblePi (talk) 15:07, 15 February 2017 (EST)