**Before I ask my question let me clarify some notation:**
$f^{i,j}\_r$, where $i < j$, refers to the inclusion map $f: H\_r(X\_i) \hookrightarrow H\_r(X\_j)$. $X\_i$ and $X\_j$ are subcomplexes of a filtered complex $X$, where $X\_i \subseteq X\_j$.


I need clarification/explanation of the following definition. I was able to understand birth events easily, but I'm having some trouble with death events. Why do the two conditions below imply the end of a certain feature?


Suppose an $r$-class $\alpha$ is born at the $i$th level. Then $\alpha$ is said to die at level $j$ iff:


* $f^{i,j}\_r(\alpha) \in \operatorname{im}(f^{i-1, j}\_r)$
* $f^{i,j-1}\_r(\alpha) \notin \operatorname{im}(f^{i-1, j-1}\_r)$


A death event then refers to the death of all homology classes in the coset $\alpha + \operatorname{im}(f^{i-1,i}\_r)$.


**The bounty is for whoever answers my question in the comments below.**



Clarification of "death event" in persistent homology
This definition of a "death event" is quite closely linked to the related definition of a "birth event", so it may be helpful to post that definition in the statement of your question as well. Also, which reference is this from?


The main idea is the following. Since $\alpha$ was born at the $i$th level, the bullet point $f\_r^{i,j-1}(\alpha)\notin \operatorname{im}(f\_r^{i-1,j-1})$ implies that $\alpha$ is still alive at index $j-1$. Roughly speaking, this is because if we suppose for a contradiction that $f\_r^{i,j-1}(\alpha)\in \operatorname{im}(f\_r^{i-1,j-1})$, then this would instead imply that $\alpha$ were born earlier (at or before index $i-1$). Additionally, the bullet point $f\_r^{i,j}(\alpha)\in \operatorname{im}(f\_r^{i-1,j})$ implies that $\alpha$ is dead by index $j$. Roughly speaking, this is because if we suppose for a contradiction that $f\_r^{i,j}(\alpha)\notin \operatorname{im}(f\_r^{i-1,j})$, then this would instead imply that $\alpha$ were still alive at index $j$. Together these two bullet points imply that $\alpha$ dies exactly at index $j$.


