Theodoros is said to have taken the next steps in the same direction by proving the irrationality of the square roots, of the non-squares from 3 to 17, but the discoverer of the irrationality of ,/ 2 was unknown to the posterity; his name has vanished and there was later a belief that he lost his life in a shipwreck as a punish- ment for his outrageous discovery, and that the mystery of irrational [...] The application of Hausdorff spaces may be considered as a geometrization; on the other hand the opportunity of deriving several theorems from one by a suitable 8choice of a topology in a set, or of the system of admissible sets, some times reminds on© of modern Algebra. [...] There will be no misinterpretation if we denote for the purpose of these lectures the set containing only one point, say a, by a. Hence a C A . (4) means that a is contained in the set A. The points contained in A &B form a set, the meet A n B . (5) of A and B; the points contained in A or B form a set, the join AUB . (6) of A and B. 1. 1. [...] Let p be a point of C not belonging to D. As D is closed, p is contained in an open set D such that [D] contains no point of D; then D fl C has the property D*; hence p is not a point of D*(C), and D*(C) = D. The class Li of all the properties corresponding to the setD contains there- fore the property D*, on the other hand, if in any subset A of C any pro- lepore)? [...] of the class A holds, the property D* holds also in A. The property D* is therefore the 'maximum' property of the class A ; it is the propeity which may be constructed by combining all the properties of the class A by or.