Here is a picture of the question, background information, and my attempt/thoughts on the problem.
My main concern, is my understanding of the distance function given. I'm not sure how to calculate the distance between two subsets of the metric space using this distance. In particular the inf, or greatest lower bound part of the definition confuses me. Some examples would really help me out.
Also, I'm not sure if the set I picked does indeed define a metric space and how to show it formally using the 3 axioms of a metric space.
Thank you!!
Best Answer
The distance $d(C,D)$ is defined to be the infimum of the set $\{d(c,d) \mid c\in C,\,d\in D\}$, which is just the set of all possible distances between elements of $C$ and elements of $D$.
For your particular example, it is true that $d((-1,0),(0,1))=0$. To see this note that $$ \{d(x,y) \mid x\in(-1,0),y\in(0,1)\}=(0,2) $$ and the infimum of this set is $0$.
Alternatively we can argue that for each $\varepsilon>0$, we have $-\varepsilon/2\in(-1,0)$ and $\varepsilon/2\in (0,1)$ so that $$ d((-1,0),(0,1)) \le d(-\varepsilon/2,\varepsilon/2) = \varepsilon. $$ Since $d((-1,0),(0,1))\le \varepsilon$ for every $\varepsilon>0$, we conclude that it is $0$.