WALKING UP TO AN L-SPACE
JUSTIN T MOORE, A SOLUTION TO THE L-SPACE PROBLEM,
PREPRINT, 2005.
Erik A. Andrejko
University of Wisconsin - Madison – SPECIALTY EXAM
November 29 2005
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

S AND L-SPACES
A space is called hereditarily separable if every subspace has a
countable dense subset.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

S AND L-SPACES
A space is called hereditarily separable if every subspace has a
countable dense subset.
A space is called hereditarily Lindelöf if every cover of a subspace
has a countable subcover.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

S AND L-SPACES
A space is called hereditarily separable if every subspace has a
countable dense subset.
A space is called hereditarily Lindelöf if every cover of a subspace
has a countable subcover.
An S-space is a regular (T3 ) space that is hereditarily separable but
not hereditarily Lindelöf.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

S AND L-SPACES
A space is called hereditarily separable if every subspace has a
countable dense subset.
A space is called hereditarily Lindelöf if every cover of a subspace
has a countable subcover.
An S-space is a regular (T3 ) space that is hereditarily separable but
not hereditarily Lindelöf.
An L-space is a regular space that is hereditarily Lindelöf but not
hereditarily separable.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

S AND L-SPACES
A space is called hereditarily separable if every subspace has a
countable dense subset.
A space is called hereditarily Lindelöf if every cover of a subspace
has a countable subcover.
An S-space is a regular (T3 ) space that is hereditarily separable but
not hereditarily Lindelöf.
An L-space is a regular space that is hereditarily Lindelöf but not
hereditarily separable.
QUESTION
(Juhász and Hajnal 1968) Do there exist S and L spaces?
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
A space X is called right-separated if it can be well ordered in type
ω1 such that every initial segment is open.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
A space X is called right-separated if it can be well ordered in type
ω1 such that every initial segment is open.
...
...
FIGURE: A right-separated space
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
A space X is called right-separated if it can be well ordered in type
ω1 such that every initial segment is open.
...
...
FIGURE: A right-separated space
A space X is called left-separated if it can be well ordered in type ω1
such that every initial segment is open.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
A space X is called right-separated if it can be well ordered in type
ω1 such that every initial segment is open.
...
...
FIGURE: A right-separated space
A space X is called left-separated if it can be well ordered in type ω1
such that every initial segment is open.
...
FIGURE: A left-separated space
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
THEOREM
A space is hereditarily separable iff it has no uncountable left
separated subspace.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
THEOREM
A space is hereditarily separable iff it has no uncountable left
separated subspace.
THEOREM
A space is hereditarily Lindelöf iff it has no uncountable right
separated subspace.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
FACT
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.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
FACT
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.
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 .
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
THEOREM
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 .
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
THEOREM
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 .
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

CANONICAL S AND L-SPACES
THEOREM
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 .
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

BUILDING S AND L-SPACES
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

BUILDING S AND L-SPACES
THEOREM (KUREPA)
A Suslin line is an L-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

BUILDING S AND L-SPACES
THEOREM (KUREPA)
A Suslin line is an L-space.
THEOREM (RUDIN)
If there is a Suslin line there is an S-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

BUILDING S AND L-SPACES
THEOREM (KUREPA)
A Suslin line is an L-space.
THEOREM (RUDIN)
If there is a Suslin line there is an S-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

BUILDING S AND L-SPACES
THEOREM (KUREPA)
A Suslin line is an L-space.
THEOREM (RUDIN)
If there is a Suslin line there is an S-space.
THEOREM
(CH) There exists S and L-spaces.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

DESTROYING S AND L-SPACES
A strong S-space is a space X such that for all n < ω X n is an
S-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

DESTROYING S AND L-SPACES
A strong S-space is a space X such that for all n < ω X n is an
S-space.
A strong L-space is a space X such that for all n < ω X n is an
L-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

DESTROYING S AND L-SPACES
A strong S-space is a space X such that for all n < ω X n is an
S-space.
A strong L-space is a space X such that for all n < ω X n is an
L-space.
THEOREM (ROITMAN)
Adding a single Cohen real adds a strong S-space and strong L-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

DESTROYING S AND L-SPACES
A strong S-space is a space X such that for all n < ω X n is an
S-space.
A strong L-space is a space X such that for all n < ω X n is an
L-space.
THEOREM (ROITMAN)
Adding a single Cohen real adds a strong S-space and strong L-space.
Under MA + ¬ CH a space is strongly hereditarily Lindelöf iff it is
strongly hereditarily separable.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

DESTROYING S AND L-SPACES
A strong S-space is a space X such that for all n < ω X n is an
S-space.
A strong L-space is a space X such that for all n < ω X n is an
L-space.
THEOREM (ROITMAN)
Adding a single Cohen real adds a strong S-space and strong L-space.
Under MA + ¬ CH a space is strongly hereditarily Lindelöf iff it is
strongly hereditarily separable.
THEOREM (KUNEN)
(MA + ¬ CH) There are no strong S or L-spaces.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

A CONJECTURE
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

A CONJECTURE
ˇ´
THEOREM (TODORCE VI C)
(PFA) There are no S-spaces
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

A CONJECTURE
ˇ´
THEOREM (TODORCE VI C)
(PFA) There are no S-spaces
CONJECTURE
(PFA) There are no L-spaces.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

PARTITIONS
The existence of S and L-spaces are equivalent to the existence of
certain colorings of [ω1 ]2 .
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

PARTITIONS
The existence of S and L-spaces are equivalent to the existence of
certain colorings of [ω1 ]2 .
THEOREM (ROITMAN)
There does not exist an L-space iff for every partition
[ω1 ]2 = K0 ∪ K1
there is an increasing (separated) sequence {aξ : ξ < ω1 } of n
elements subsets of ω1 and a k < n such that for all ξ < η either
(I) There is an i > k such that {aξ i , aηk } ∈ K1 ,
(II) or for some i ≤ k {aξ i , aηk } ∈ K1 iff {aξ i , aξ k } ∈ K0
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
... ...
FIGURE: Tr(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
... ...
FIGURE: Tr(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
...
... ...
FIGURE: Tr(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
... ...
... ...
FIGURE: Tr(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
... ...
... ...
...
FIGURE: Tr(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

UPPER TRACE
Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The
sequence Cδ : δ < ω1 is called a ladder system which allows on to
\"walk\" from β to α.
Define the upper trace Tr(α, β )
Tr(α, α) = 0,
/
Tr(α, β ) = Tr(α, min(Cβ \\ α)) ∪ {β }
... ...
... ...
...
FIGURE: Tr(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
... ...
FIGURE: L(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
... ...
FIGURE: L(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
...
... ...
FIGURE: L(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
... ...
... ...
0
FIGURE: L(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
... ...
... ...
...
01
FIGURE: L(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

LOWER TRACE
Define the lower trace L(α, β ) by
L(α, α) = 0,
/
L(α, β ) = L(α, min(Cβ \\ α)) ∪ {max(Cβ ∩ α)} \\ max(Cβ ∩ α)
... ...
... ...
...
01
2
FIGURE: L(α, β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

COHERENT SEQUENCES
If eβ : β < ω1 is a sequence of finite to one functions such that
eβ → ω for all β < ω1 then it is a coherent sequence if whenever
β ≤ β eβ β differs from eβ on a finite set.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

COHERENT SEQUENCES
If eβ : β < ω1 is a sequence of finite to one functions such that
eβ → ω for all β < ω1 then it is a coherent sequence if whenever
β ≤ β eβ β differs from eβ on a finite set.
For α ≤ β define ρ1 (α, β ):
max |Cξ ∩ α|
ρ1 (α, β ) =
ξ ∈Tr(α,β )
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

COHERENT SEQUENCES
If eβ : β < ω1 is a sequence of finite to one functions such that
eβ → ω for all β < ω1 then it is a coherent sequence if whenever
β ≤ β eβ β differs from eβ on a finite set.
For α ≤ β define ρ1 (α, β ):
max |Cξ ∩ α|
ρ1 (α, β ) =
ξ ∈Tr(α,β )
Define eβ (α) = ρ1 (α, β ).
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

COHERENT SEQUENCES
If eβ : β < ω1 is a sequence of finite to one functions such that
eβ → ω for all β < ω1 then it is a coherent sequence if whenever
β ≤ β eβ β differs from eβ on a finite set.
For α ≤ β define ρ1 (α, β ):
max |Cξ ∩ α|
ρ1 (α, β ) =
ξ ∈Tr(α,β )
Define eβ (α) = ρ1 (α, β ).
FACT
The sequence eβ : β < ω1 is a coherent sequence.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OSCILLATIONS
For finite functions s and t defined on a common set F let Osc(s, t; F )
be the set of all ξ ∈ F \\ {min(F )} such that s(ξ − ) ≤ t(ξ − ) and
s(ξ ) > t(ξ ) where ξ − is greatest element of F less than ξ .
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OSCILLATIONS
For finite functions s and t defined on a common set F let Osc(s, t; F )
be the set of all ξ ∈ F \\ {min(F )} such that s(ξ − ) ≤ t(ξ − ) and
s(ξ ) > t(ξ ) where ξ − is greatest element of F less than ξ .
Define osc(α, β ) by:
osc(α, β ) = | Osc(α, β ; L(α, β ))|
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OSCILLATIONS
THEOREM (MOORE)
For any A ⊆ [ω1 ]k and B ⊆ [ω1 ]l which are pairwise disjoint and
uncountable and every n < ω there exist a ∈ A and {bm }m<n ⊆ B with
a < bm and for all i < k, j < l and m < n
osc(a(i), bm (j)) = osc(a(i), b0 (j)) + m
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

THE MAIN RESULT
THEOREM (MOORE)
There exists a function o∗ : ω1 → ω such that if A ⊆ [ω1 ]k and
2
B ⊆ [ω1 ] are pairwise disjoint and uncountable and χ : k → ω is any
l
function then there exist a ∈ A and b ∈ B with a < b and
o∗ (a(i), b(φ (i))) = χ(i)
for any φ : k → l.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

THE MAIN RESULT CONT...
COROLLARY
There exists an L-space.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

THE MAIN RESULT CONT...
COROLLARY
There exists an L-space.
COROLLARY
There exists a weak HFC.
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OPEN QUESTIONS
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OPEN QUESTIONS
QUESTION
Does there exist an L-space whose square is an L-space?
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OPEN QUESTIONS
QUESTION
Does there exist an L-space whose square is an L-space?
QUESTION
Does there exist an HFCk space?
w
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

OPEN QUESTIONS
QUESTION
Does there exist an L-space whose square is an L-space?
QUESTION
Does there exist an HFCk space?
w
QUESTION
Does there exist an L-space that is not a weak HFC?
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE

REFERENCES
ERIK A. ANDREJKO WALKING UP TO AN L-SPACE