The concept of a Schauder basis is central to Banach space theory, but occasionally too wide to be useful in Banach lattices, while too restrictive for general separable Banach spaces. In Banach lattices, new notions of bases incorporate different types of order convergence, whereas in Banach space, one can conversely weaken sequential convergence to filter convergence. Applying tools from descriptive set theory, we answer several fundamental questions due to Gumenchuk, Karlova and Popov (J. Funct. Anal. 2015), Taylor and Troitsky (J. Funct. Anal. 2020), and de Rancourt, Kania and Swaczyna (J. Funct. Anal. 2023) related to these concepts. This is a joint work with A. Avilés, M. A. Taylor and P. Tradacete.