By Wolfgang Lück

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2, 65-77 (1937); 6, 1-17 (1941); 7, 65-89, 133-145 (1942); 8, 89-106 (1943); 13, 65-79 ( 1948) ; 19, 8 1-96 ( 1954). 4. K. Giidel, “ T h e Consistency of the Axiom o f Choice iind o f the Generalized Continuum-Hypothesis with the Axioms of Set ‘llicory” (Ann. Math. Studies, No. 3) Princeton Univ. Press, Princeton, New Jersey. 1940. 5 . E. Zermelo, Be\veis dass jede l l e n g e wohlgeordnct iverden kann. Mn/h. Ann. 59, 514-516 (1904). 6. E. Zernielo, Keuer Ueweis f u r die Rloglichkeit eiiic’r \\’ohlordnung.

Is an open set containing x, then O,rn Bi is open and contains x, so by x E A it intersects A. Therefore O,rn B n A is not void and s o x E A n B. Hence if x E ,q n B', then x E A n B n B'. Proof. EXERCISES I . Show that 0 is open if and only if A n0 = 0 implies An0 = 0 for every '4 in '71. 2. I n Ri 0. (First, if x c$ '4, then there is an open set 0, containing x such A n O,r = 0. x E B'. we have 0, n A 3. Determine A ' , A", A'), A, A', and for every set A when X is topologized by the topology of finite complements.

53, I 10-1 I 3 (1950). 10. N . ” Part I . 1,ivre I . Theorie des ensembles (Fascicule de resultats). (Actual. Sci. , no. ) Herrnann, Paris, 1939. This Page Intentionally Left Blank CHAPTER I Topological Spaces 1. Open Sets and Closed Sets We shall deal with mathematical systems which consist of a set X and a family of subsets of X which are subject to a few simple axioms. These systems are called topological spaces and the family 0, of subsets is the family of open sets. Definition 1. 3) 0~ 0 If O , E O and /f OiE U for ewery and X E 0.