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Lattices and topologies

Lattices and topologies are indispensable tools for working logicians. On the one hand, lattices are algebraic structures describing behavior of such basic logical connectives as conjunction and disjunction. On the other hand, representation theorems for lattices in terms of certain (ordered) topological spaces can be thought of as completeness results for several important logical systems. It is our intention to present a systematic study of basics of lattices and topologies and their connection. We describe the Stone duality between distributive lattices and spectral spaces, and its equivalent formulation in terms of Priestley spaces and bitopological spaces. We conclude by explaining relationship between the resulting representation theorems and topological completeness theorems in logic. Prerequisites: introductory knowledge of set theory.