Title: Relational T-algebra and the category of topological spaces

Abstract: 
  Continuous lattices are used in computer science such as a denotational semantics of a program language. A. Day showed an equivalence between the category of T-algebra defined by the filter monad and the category of continuous lattices and continuous functions. Our motivation is developing a formal algebraic theory for a mathematical objects which is used in computer science.
  M. Barr introduced a notion of relational T-algebra which is an extension of T-algebra. He also showed an equivalence between the category of relational T-algebra defined by the ultra-filter monad and the category of topological spaces and continuous functions. We refine their theories and introduce a formal proof using  relational calculus.