  McKinsey and Tarski initiated the study of interior algebras. We propose
complete interior algebras as an alternative pointfree approach to topology. We
term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating
with each MT-algebra the lattice of its open elements defines a functor from
the category of MT-algebras to the category of frames, which we study in depth.
We also study the dual adjunction between the categories of MT-algebras and
topological spaces, and show that MT-algebras provide a faithful generalization
of topological spaces. Our main emphasis is on developing a unified approach to
separation axioms in the language of MT-algebras, which generalizes separation
axioms for both topological spaces and frames.
