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

1. Basic S and L
The Existence of an S-space under MA + ¬CH
Erik A. Andrejko
May 25, 2007
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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?
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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.
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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.
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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 .
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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.
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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?
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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.
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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.
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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.
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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.
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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)
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