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Basic S and L : The existence of an S-space under MA and \$\neg\$ CH

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Basic S and L : The existence of an S-space under MA and \$\neg\$ CH

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Basic S and L : The existence of an S-space under MA and \$\neg\$ CH

1. 1. Basic S and L The Existence of an S-space under MA + ¬CH Erik A. Andrejko May 25, 2007 1
2. 2. S and L spaces The study of S and L spaces is one of the most active areas of research in set theoretic topology. The notions of an S space and L space are in some sense duals of each other and many of their existance properties coincide. A space is called hereditarily separable if every subspace has a countable dense subset. A space is called hereditarily Lindel¨f if every cover of a o subspace has a countable subcover. An S-space is a regular (T3 ) space that is hereditarily separable but not hereditarily Lindel¨f. An L-space is a regular space that is hereditarily Lindel¨f o o but not hereditarily separable. If you drop regular than both spaces exists as reﬁnements of the topology on a well ordered subset of R of type ω1 . The basic question: when do such spaces exist? Do they always coincide? 2
3. 3. Canonical S and L-spaces A space X is called right-separated if it can be well ordered such that every initial segment is open. If the order type of X is at least ω1 then X cannot be Lindel¨f. A space X is called left-separated if it can be well ordered such that o every initial segment is open. If the order type of X is at least ω1 then it cannot be separable. ... Figure 1: A right-separated space ... Figure 2: A left-separated space Theorem 1. A space is hereditarily separable iﬀ it has no uncountable left separated subspace. A space is hereditarily Lindel¨f iﬀ it has no uncountable right o separated subspace. 3
4. 4. Proof. The proof of one direction is obvious. For the other direction, if a space is not hereditarily Lindel¨f there is a strictly increasing sequence of open sets o {uα : α < ω1 }. We can then construct a right separated subspace by choosing xα ∈ uα+1 uα . If Y is a non separable subspace then we can deﬁne a sequence of points {xα : α < ω1 } with xα ∈ cl{xβ : β < α}. Then each initial segment of / {xα : α < ω1 } is closed and so the subspace is left-separated of type ω1 . Fact 1. A regular right separated space of type ω1 is an S-space iﬀ it has no uncountable discrete subspace. A regular left separated space of type ω1 is an L-space iﬀ it has no uncountable discrete subspaces. 4
5. 5. By 2ω1 we denote the Tychanoﬀ product space where 2 is the 2 point discrete space. This has basis elements [σ] where σ is a ﬁnite function from ω1 to 2 and where [σ] = {f ∈ 2ω1 : f|dom(σ) = σ} i.e., the functions which extend σ. Note that such a space is automatically T3 since the topology on 2 is discrete and the product of T3 spaces is T3 . In fact each basic open neighborhood [σ] is automatically closed. ... ... Figure 3: A basic open neighborhood [σ] and f1 , f2 ∈ [σ] Theorem 2. Canonical Form (a) Every S-space contains an uncountable subset which under a possibly weaker topology is homeomorphic to a right separated S-subspace of 2ω1 . (b) Assume ¬CH. Every L-space contains a subset which under a possibly weaker topology is homeomorphic to a left separated L-subspace of 2ω1 . Thus S and L-spaces, when they exist, exist as subspaces of 2ω1 . 5
6. 6. Building S and L-spaces It is possible to construct both S and L-spaces under CH. A subset X ⊆ 2ω1 is called a ﬁnally dense if for some α ∈ ω1 for all σ : ω1 α → 2, σ a ﬁnite function, then [σ] ∩ X is inﬁnite. A set X is called hereditarily ﬁnally dense, or a HFD if every inﬁnite subset of X is ﬁnally dense. Fact 2. HFD’s are hereditarily separable. Fact 3. If there exists an HFD then there exists an S-space. Theorem 3. (CH) There exists an HFD. To construct an L-space, one starts with a subset X ⊆ 2ω1 called an HFC. Similarly as with HFD’s the following facts are true: Fact 4. HFC’s are hereditarily Lindel¨f. o Fact 5. If there exists an HFC then there exists an L-space. Theorem 4. (CH) There exists an HFC. Combining the previous two theorems gives the following existence theorem. Theorem 5. (CH) There exists an S-space and an L-space. 6
7. 7. Destroying S and L-spaces As easily as S and L-spaces exist under CH they can be destroyed under MA + ¬CH. A strong S-space is a space X such that for all n < ω Xn is an S-space. A strong L-space is a space X such that for all n < ω Xn is an L-space. The following are theorems under MA + ¬CH: Theorem 6. (Kunen) There are no strong S or L spaces. Theorem 7. (Juh´sz) There are no compact L-spaces. a Theorem 8. (Szentmikl´sssy) There are no compact S-spaces. o Theorem 9. (Szentmikl´sssy) There are no ﬁrst countable L-spaces. o The natural question is then, do S and L-spaces exist under MA + ¬CH? 7
8. 8. The Existence of an S-space under MA + ¬CH ccc forcing M |= CH +3 M[G] |= MA + ¬CH THFD X / THFD X S-space X The proof of the existence of an S-space under MA + ¬CH proceeds as follows: (i) Under CH construct a THFD X. (ii) Show X is ccc-indestructible. (iii) Do a ccc-forcing to get an M[G] |= MA + ¬CH. (iv) In M[G] prove that X is an S-space. Then M[G] |= MA + ¬CH + ∃ an S-space. 8
9. 9. The Existence of an S-space under MA + ¬CH Let A have order type ω. Then B ⊆ A is tight in A iﬀ for some n ω every set of n consecutive members of A contains a member of B. Let X = {xα : α ω1 } ⊆ 2ω1 . X is a THFD iﬀ for every countable set A ⊆ X or limit type there is some α ω1 such that for all σ : ω1 α → 2 a ﬁnite function then A ∩ [σ] is tight in A. That is, for some n, for every n consecutive members of A one must extend σ. Theorem 10. Assume CH. Then there is a THFD. Since a THFD is an HFD then the existence of an HFD proves the existence of an S-space. To complete the proof of the consistency of the existence of an S-space under MA + ¬CH it remains to show that a THFD is ccc-indestructible. 9
10. 10. THFD’s are ccc-indestructible Let X be a space. The partial order PX is the set of maps from ﬁnite subsets of X to open sets of X such that for all p ∈ PX if x = y then p(x) = p(y). Order PX by reverse inclusion. The condition p forces that dom(p) is discrete. If G is a ﬁlter in PX then p∈G dom(p) is discrete. Figure 4: A forcing element p ∈ PX ¯ For A ⊆ PX deﬁne A to be the closure of A under ﬁnite unions of mutually compatible conditions. Let M be a model of ZFC, then for X ∈ M we say that X is ccc-indestructible in M if for any ccc Q-generic G, M[G] |= X has no uncountable discrete subspaces. Lemma 1. Let P be a ccc partial order. If P is uncountable then there is some A ∈ [P]ω1 such that for all p ∈ A there exists uncountably many r ∈ A so that p ⊥ r. Theorem 11. If X is THFD then X is ccc-indestructible. 10
11. 11. Some Further Results Theorem 12. (Abraham, Todorˇevi´) [AT84] The following is consistent: cc MA + ¬CH+ there exists a ﬁrst countable S-space. Thus, under MA + ¬CH there is a ﬁrst countable S-space but no ﬁrst countable L-space. The duality between S and L-spaces breaks down further. Theorem 13. (Baumgartner) MA + ¬CH + TOP implies there are no S-spaces. Thus it is consistent that there are no S-spaces. It was unknown if it was consistent that there are no L-spaces until this year when it was proven: Theorem 14. (Moore) [Moo05] There exists an L-space. Which is a theoreom of ZFC only and does not require any additional axioms. This answers the question that the consistence of there does not exist an L-space is false. 11
12. 12. References [AT84] U. Abraham and S. Todorˇevi´, Martin’s axiom and ﬁrst-countable S- cc and L-spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 327–346. MR MR776627 (86h:03092) [Moo05] Justin T Moore, A solution to the l space problem and related zfc constructions, preprint, 2005. [Roi84] Judy Roitman, Basic S and L, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 295–326. MR MR776626 (87a:54043) 12