# Property of Hausdorff distance on set minus

I am trying to understand how the Hausdorff distance acts on complements but struggling to find any good resources. Is it true in general that if I have $$3$$ compact sets $$A,B$$ and $$C$$ that the following implication holds?

$$d_H(A,B)\leq r \implies d_H(C\setminus A, C\setminus B) \leq r$$

Where $$d_H$$ is the Hausdorff distance (assume the sets $$C\setminus A$$ and $$C\setminus B$$ are non-empty). Also, does anyone know of any good resources for further reading on the Hausdorff distance? Any with exercises would really help!

## 1 Answer

Update: I think I have a counterexample in $$\mathbb{R}^2$$:

Let

$$A = \bigg(\bigg[\frac{1}{n},1\bigg]\times [0,1]\bigg) \cup ([0,1]\times [2,3])$$ $$B = ([0,1]\times [0,1]) \cup ([0,1]\times [2,3])$$ $$C = ([0,1]\times [0,1]) \cup ([0,1]\times [1,3])$$

We have that:

$$C\setminus A =\bigg(\bigg[0,\frac{1}{n}\bigg)\times [0,1]\bigg) \cup ([0,1]\times [1,2))$$ and:

$$C\setminus B =[0,1]\times [1,2)$$

Trivially we can make $$d_H(A,B)$$ as small as we like by increasing $$n$$. But for $$d_H(C\setminus A,C\setminus B)$$ will always be greater than or equal to $$1$$.