By Dixon P.

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ISBN 3-540-08520-3. ] 21. , ISBN 3-540-13258-9. ] 22. , ISBN 3-540-065229. ] 23. Andras Hajnal and Peter Hamburger, ‘Set Theory’, (CUP, LMS Student Texts, 48, 1999), 316pp, ISBN 0-521-59667-X.

10 (a) (ZF) A topological space is Hausdorff if and only if no filter converges to more than one point. (b) (ZF)+(UFT) A topological space is compact if and only if every ultrafilter converges. (c) (ZF) A function f : X → Y is continuous if and only if f (F) → f (x) whenever F → x. Proof. The proofs of (a) and (c) are straightforward; we prove (b). Suppose that X is compact and that U is an ultrafilter on X. Let U be the set of all closures of members of U. Then, since U has the finite intersection property, so does U .

Let κ = cf (α) and let γ = max{β, κ}. Then 2ℵα = 2< ℵα κ, by the lemma, = 2ℵγ κ, since γ > β, = 2ℵγ , since γ > κ, = 2ℵβ , since γ > β. ♦ Another constraint on the continuum function is the following theorem of Silver8 . 20 Let ℵα be a singular cardinal with cf (α) > ω. e. 2ℵβ = ℵβ+1 (β < α) ⇒ 2ℵα = ℵα+1 . 6 L. Bukovsky, ‘The continuum problem and the powers of alephs’, Comment. Math. Univ. Carolinae, 6 (1965), 181– 197. H. Hechler, ‘Powers of singular cardinals and a strong form of the negation of the generalized continuum hypothesis’, Z.