a note on approximation conditions, standard triangularizability and a power set topology

the main result of this article is that for collections of entry-wise non-negative matrices the property of possessing a standard triangularization is stable under approximation. the methodology introduced to prove this result allows us to offer quick proofs of the corresponding results of [b. r. yahaghi, near triangularizability implies triangularizability, canad. math. bull. 47, (2004), no. 2, 298--313], and [a. a. jafarian, h. radjavi, p. rosenthal and a. r. sourour, simultaneous, triangularizability, near commutativity and rota's theorem, trans. amer. math. soc.  347, (1995), no. 6, 2191--2199] on the approximations and triangularizability of collections of operators and matrices. in conclusion we introduce and explore a related topology on the power sets of metric spaces.