Notation 
 This page uses non-standard notation filtration and homology, maybe following Zomorodian's book. Across wikipedia, and in Edelsbrunner and Harel's book, the filtration indices and homology group indices are both represented by subscripts. Since the author here didn't define the notation they are using, I'm moving the existing definition over here and giving the one from E&H;, p. 151. 

 Let  be a filtration. The p-persistent kth homology group of  is .  

 If we let  be a nonbounding -cycle created at time  by simplex  and let   be a homologous -cycle that becomes a boundary cycle at time  by simplex , then we can define the persistence interval associated to  as . We call  the creator of  and  the destroyer of . If  does not have a destroyer, its persistence is . 

 Instead of using an index-based filtration, we can use a time-based filtration. Let  be a simplicial complex and  be a filtration defined for an associated map  that maps simplices in the final complex to real numbers. Then for all real numbers , the -persistent kth homology group of  is . The persistence of a -cycle created at time  and destroyed at  is . 
 Afra J. Zomorodian (2005): Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics.