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, ‎\textit{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 Rotaʹs theorem‎, ‎\textit{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.