Consider $X:= (0, 1], Y:=[-1, 0)$. I'm trying to verify that $d_H(X, Y)=1$ (as stated in https://en.wikipedia.org/wiki/Hausdorff_distance) using the definition of $d_H$ with the $\delta$-neighbourhoods..
I tried to show a double inequality. The first one is clear, but I have some problems to prove that $d_H(A, B) \geq 1$.
Probably it's a basic question, but I'm new with this concept/definition, so any hint/help would be appreciate.
Thanks in advance!
It remains to show, that $d_H (X,Y) \ge 1$. Using the definition you mentioned, to show that this infimum is at least $1$ we have to show, that either $X \not\subset Y_\varepsilon$ or $Y \not\subset X_\varepsilon$ for all $0<\varepsilon<1$. For the sets from your question we have explicitly:$$X_\varepsilon = [-\varepsilon,1+\varepsilon].$$We therefore see that for any $0 < \varepsilon <1$ we have $Y \not\subset X_\varepsilon$. Hence $d_H (X,Y) \ge1$.