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.