By Thomas Jech (auth.)

Set thought has skilled a speedy improvement lately, with significant advances in forcing, internal types, huge cardinals and descriptive set conception. the current ebook covers every one of those parts, giving the reader an knowing of the tips concerned. it may be used for introductory scholars and is large and deep sufficient to carry the reader close to the limits of present examine. scholars and researchers within the box will locate the ebook priceless either as a examine fabric and as a laptop reference.

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Proof. Let κ = |S| and λ = sup{|X| : X ∈ S}. We have S = {Xα : α < κ} and for each α < κ, we choose an enumeration Xα = {aα,β : β < λα }, where λα ≤ λ. Again we have a projection (α, β) → aα,β of κ × λ onto S, and so | S| ≤ κ · λ. In particular, the union of ℵα sets, each of cardinality ℵα , has cardinality ℵα . 3. Every ℵα+1 is a regular cardinal. Proof. This is because otherwise ωα+1 would be the union of at most ℵα sets of cardinality at most ℵα . Using the Axiom of Choice in Mathematics In algebra and point set topology, one often uses the following version of the Axiom of Choice.

Let f : R → R. Show that the set of all x at which f is continuous is a Gδ set. 17. (i) N × N is homeomorphic to N . (ii) N ω is homeomorphic to N . 18. , s ∈ T such that there is no t ∈ T with s ⊂ t. The map F → TF is a one-to-one correspondence between closed sets in N and sequential trees without maximal nodes. 19. Every perfect Polish space has a closed subset homeomorphic to the Cantor space. 20. Every Polish space is homeomorphic to a Gδ subspace of the Hilbert cube. 5 are due to Cantor. The construction of real numbers by completion of the rationals is due to Dedekind [1872].

N ∈ N , are dense open sets of reals, then the intersection D = n=0 Dn is dense in R. Proof. We show that D intersects every nonempty open interval I. First note that for each n, D0 ∩ . . ∩ Dn is dense and open. Let Jk : k ∈ N be an enumeration of rational intervals. Let I0 = I, and let, for each n, In+1 = Jk = (qk , rk ), where k is the least k such that the closed interval [qk , rk ] is included in In ∩ Dn . Then a ∈ D ∩ I, where a = limk→∞ qk . 9. 2) (i) S ∈ S, (ii) if X ∈ S and Y ∈ S then X ∪ Y ∈ S, (iii) if X ∈ S then S − X ∈ S.

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