Does Adding the Boundary of a Set A Affect Its Distance Metric?

[yifli]

Homework Statement

show that \rho(x,A)=\rho(x,\bar{A}), where \rho is a distance metric

Homework Equations

\rho(x,A)=glb\left\{\rho(x,\alpha),\alpha \in A \right\}
\bar{A} is the closure of A
\partial A, the boundary of an arbitrary set A is the difference between its closure and its interior

The Attempt at a Solution

If A is a closed set, then A=\bar{A}
If A is open, I want to prove adding boundary element of A has no impact on \rho(x, A). this seems to be intuitive, but can't come up with a rigorous proof.

 

[henry_m]

My initial reaction is that proof by contradiction will be your best bet. It's clear that \rho(x,A)\geq\rho(x, \bar{A}) since A\subseteq\overline{A}, so assume that the inequality is strict, and try to derive a contradiction.