The notion of Galois Connections between partially ordered sets is introduced, together with a presentation of some of its main characterizations. This leads to a true understanding of the deep connection that links Galois Connections to Quillen’s Homotopy Type Equivalence Theorem. Furthermore, the notion of Möbius functions of finite lattices is discussed, in order to show its crucial role in Enumerative Combinatorics over Finite Posets and Discrete Probability Theory. Since the values of the Möbius function of a finite lattice may be regarded as reduced Euler Characteristic of suitable topological spaces, a wide variety of combinatorial results can be proved by topological methods. We exploit this point of view by providing elementary proofs of two classical theorems: the «Cross-Cut Theorem» of Rota and the «Vanishing Theorem for not-strongly complemented lattices» of Crapo.