By K. Kuratowski (Auth.)

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It follows by (23) and (16), X€A%^}mmk which is a con tradiction to (22). Thus (18) and (19) have been established. I t now only remains to show (14). To this end, in view of (18) and (17), we have to prove that a a But this follows from (19), because if xeEa for each a, then there is an a such that xe(Ea-Ta) =EaczU α<Ω Ka. *XV. Lusin sieve(x)· Let ^ 0 be the set of binary fractions r = 1 1 "2^Γ+···+ 2 ^ Where 1 < m i<---< m n. (1) A mapping W which assigns to each r c ^ 0 a s e t f r c l (where X is a fixed set) is called a sieve.

Szpilrajn-Marczewski, Fund. Math. 26(1936), p . 302, and 31 (1938), p. 207. [§ 3 ] MAPPINGS. ORDERINGS. CARDINAL AND ORDINAL NUMBERS 27 of X onto Ύ such that (xx^x2) ^(f(x1)<*f(x2)). The relation between two ordered sets making them similar is an equivalence relation. Consequently, one can assign order types to ordered sets so that the same order type is assigned to two ordered sets if and only if they are similar (this procedure is comple tely analogous to the procedure of assigning to sets their cardinal numbers).

335, and Topologia (1928). See also § 7, I I below. Compare A. Monteiro, Les ensembles fermés et les fondements de la Topologie, Portug. Math. 2 (1941), pp. 56-66. 46 TOPOLOGICAL SPACES [CH. I Proof. The condition is sufficient, for if X is relatively closed, then X = E n X, and if X is relatively open, then X = E—E—X = E n(l-E-X). To show that the condition is sufficient, suppose first that X — E n F and F = F. e. that X = E n X or that E nF = E n E n F. Thus, by rule 1, E r\ F a F and since F = F, then EnEnFaEn F.