Say we have an essential class (barcode $[a,\infty)$, meaning its represents a feature that never gets killed) represented in a persistence diagram on $\bar{\mathbb{R}}^2$ as x= $(a,\infty)$. Where $\bar{\mathbb{R}}= \mathbb{R}\cup \{\infty\}$
What is x= $(a,\infty)$'s $L_\infty$ distance to $(b,\infty)$ and (b,c) $a,b,c <\infty$?
where $\vert (u,v)-(x,y)\vert_\infty$= max(|u-x|, |v-y|)
I believe the $L_{\infty}$ distance between $(a,\infty)$ and $(b,\infty)$ is |a-b| however I do not understand why we can ignore $\infty$.
My understanding is the same as yours: the $L_\infty$ distance from $(a,\infty)$ to $(b,\infty)$ is $|a-b|$, and this follows since the "distance" between $\infty$ and $\infty$ in $\overline{R}$ is defined to be zero. Similarly, for $c<\infty$, the $L_\infty$ distance from $(a,\infty)$ to $(b,c)$ is $\infty$, and this follows since the "distance" between $\infty$ and $c$ in $\overline{R}$ is defined to be $\infty$.