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Finite topological spaces have been of little concern to topologists who are primarily interested in the properties of continua. Consequently, such mathematical structures have received scant attention in the literature. A study has found, however, that finite spaces have a rich combinatorial structure, and much of this book is devoted to developing the mathematical machinery to analyze this structure. The book is in two parts: Part 1 introduces the mathematics of finite topological spaces, and part 2 describes some applications of the results of Part 1 to questions of molecular structure, specifically topological measures of molecular complexity and of bond-strength patterns in unsaturated pi-electron systems. Some preliminary results on the topological analysis of complex reaction networks are also presented.