Acta Mathematica Sinica, New Series
1997, January, Vol.13, No.1, pp. 59–63
The Space of Strong Types and
an Open Map Theorem
Shen Yunfu
(Department of Mathematics, Beijing Normal University, Beijing 100875, China)
Abstract We define a new topology on the space of strong types on a subset A of a model of a
given theory and prove that either |Sp∗ (A)| < ℵ0 or |Sp∗ (A)| ≥ 2ℵ0 . We also deduce an open map
theorem.
Keywords Strong type, Open map, Model theory
1991 MR Subject Classification 03C
Chinese Library Classification O141.4
In [2] the authors defined a new topology on the space of strong types of a given theory
and proved an omitting types theorem for countably saturated models of the theory. As a
continuation, we now discuss the space of strong types and its topology on a subset of a model
of the theory.
1
The Space of Strong Types
Let T be a (possibly incomplete) theory in a countable language with no finite models, M
a monster model of T . We assume that all models and sets that we deal with are contained in
M.
For A ⊂ M , we denote by Sn (A) the set of all n-types on A and by Stn (A) the set of all
strong types on A. Denote S(A) = n<ω Sn (A), St(A) = n<ω Stn (A). Let τn (A) denote
the usual Stone topology on Sn (A), τ (A) the disjoint union topology of τn (A) (n < ω) on S(A).
Let (Di ; i < λ) list all equivalence classes of all A-definable finite equivalence relations on
M of arbitrary arity. Let L∗A = LA ∪ {Ui : i < ω}, where Ui is a new predicate symbol
corresponding to Di , and let MA∗ = (MA , Ui )i<λ , TA∗ = T h(MA∗ ), where Ui is interpreted by Di .
It is easily seen that MA∗ is still saturated and that for any L∗A -formula φ(x̄) there is a finite
I ⊆ λ such that
Ui (x̄)).
(∗)
TA∗ |= ∀x̄(φ(x̄) ↔
i∈I
We denote by Sn∗ (A) the set of all n-types of TA∗ on A, S ∗ (A) = n<ω Sn∗ (A). For any
p(x̄) ∈ Sn (A), let Sp∗ (A) = {q ∈ Sn∗ (A) : p ⊆ q}. Let τp denote the usual Stone topology
on Sp∗ (A), and let τn∗ (A) and τ ∗ (A) denote the disjoint union topologies on Sn∗ (A) and S ∗ (A)
respectively.
Received November 2, 1994, Accepted June 23, 1995
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From (∗) we see that τ ∗ (A) is generated by the sets of the form {q ∈ S ∗ (A) : p ∪ {Ui } ⊂
q}, p ∈ S(A). Let Ui p = {q ∈ S ∗ (A) : p ∪ {Ui } ⊂ q}. Denote by t∗ (ā/A) the L∗A -type
determined by ā on A and by F E n (A) the set of all A-definable finite equivalence relations of
n-ary on M.
Obviously, for any ā, b̄ ∈ M,
St(ā/A) = St(b̄/A) iff t∗ (ā/A) = t∗ (ā/A).
In the following we assume that T is complete.
Proposition 1 For p ∈ Sn (A), Sp∗ (A) is a compact space and all closed open sets of Sp∗ (A)
are of the form φp , where φ is an n-ary formula of L∗A .
Lemma 1 Let q ∈ Sp∗ (A). If {q} is nowhere dense (with respect to τp ) then |Sp∗ (A)| ≥ ℵ0 .
Proof Suppose {q} is nowhere dense. Then for any φp q, there is a q ∈ φp such that
q = q.
Choose φ0 ∈ q. Then there is a q0 ∈ φ0 p such that q0 = q. Let φ1 ∈ q \ q0 and denote
/ φ1 p . Hence there is a q1 ∈< φ1 >p such that q1 = q.
φ1 = φ0 ∧ φ1 . Then q ∈ φ1 p , q0 ∈
Choose φ2 ∈ q \ q1 and denote φ2 = φ1 ∧ φ2 , and so on. So we can find qn ∈ Sp∗ (A), φn , n < ω
such that
(1).
φn+1 → φn
(2).
q ∈ φn p ,
qn ∈ φn p \ φn+1 p .
(1)
Then it is easily seen that qi = qj , (i = j). Thus |Sp∗ (A)| ≥ ℵ0 .
Lemma 2 If, for any q ∈ Sp∗ (A), {q} is not nowhere dense, then there exists an E(x̄, ȳ) ∈
F E(A) such that
m
Ui p , (2). |Ui p | ≤ 1,
(1).
Sp∗ (A) =
i=1
where U1 , ..., Um are the predicates corresponding to D1 , ..., Dm which are all of the equivalence
classes for E(x̄, ȳ).
Proof
Since Sp∗ (A) is compact and any q ∈ Sp∗ (A) is isolated, we assume Sp∗ (A) =
m
i=1 Uki p and Uki p contains only one type.
Choose Ei (x̄, ȳ) such that an equivalence class of Ei corresponds to Uki .
Now suppose there does not exist such E(x̄, ȳ) as mentioned in the lemma. For any E(x̄, ȳ)
we have (1) but not (2). Then for E1 there are t∗ (ā1 /A), t∗ (b̄1 /A) ∈ Sp∗ (A) such that t∗ (ā1 /A) =
t∗ (b̄1 /A), E1 (ā1 , b̄1 ).
Denote j0 = 1 and E 0 = Ej0 , Then there is a j1 ∈ {1, ..., m} such that Ukj1 (ā1 ), ¬Ukj1 (b̄1 ).
That is, ¬Ej1 (ā1 , b̄1 ).
Denote E 1 = E 0 ∧ Ej1 . There are t∗ (ā2 /A), t∗ (b̄2 /A) ∈ Sp∗ (A), t∗ (ā2 /A) = t∗ (b̄2 /A), E 1 (ā2 ,
b̄2 ). As above, there is a j2 ∈ {1, ..., m} such that ¬Ej2 (ā2 , b̄2 ). Denote E 2 = E 1 ∧ Ej2 .
In general, there are j2 , ..., jt ∈ {1, ..., m} and ā2 , b̄2 , ..., āt , b̄t ∈ M such that
E k−1 (āk , b̄k ),
¬E k (āk , b̄k ),
E k = E k−1 ∧ Ejk .
Now it is easily seen that if t > m there are l < k ≤ t such that jl = jk . Note that E l (āk , b̄k ).
So Ejl (āk , b̄k ). That is, Ejk (āk , b̄k ). But this is a contradiction.
Corollary |Sp∗ (A)| ≥ ℵ0 iff there is a q ∈ Sp∗ (A) such that {q} is nowhere dense.
Theorem 1 If |Sp∗ (A)| ≥ ℵ0 , then |Sp∗ (A)| ≥ 2ℵ0 .
Shen Yunfu
The Space of Strong Types and an Open Map Theorem
61
Proof Suppose |Sp∗ (A)| ≥ ℵ0 . Then there is no such E(x̄, ȳ) ∈ F E(A) as required in
Lemma 2. Hence for any E ∈ F E n (A), there are t∗ (b̄/A), t∗ (c̄/A) ∈ Sp∗ (A) and an E (x̄, ȳ) ∈
F E(A) with ¬E (b̄, c̄), E(b̄, c̄).
Let ā be any realization of p. By p = t(ā/A) = t(b̄/A) there is an A−isomorphism f
satisfying f (b̄) = ā. Since E(b̄, c̄) ∧ ¬E (b̄, c̄), E(ā, f (c̄)) ∧ ¬E (ā, f (c̄)).
Let ā0 = f (c̄). p = t(ā0 /A). As above, there is an ā such that |= E(ā0 , ā ) ∧ ¬E (ā0 , ā ),
p = t(ā /A). Hence
|= E(ā, ā0 ) ∧ E(ā, ā ) ∧ ¬E (ā0 , ā ),
t∗ (ā /A),
t∗ (ā0 /A),
t∗ (ā /A) ∈ Sp∗ (A).
Then there are U (x), U0 (x), U1 (x) such that
t∗ (ā/A) ∈ U p ,
t∗ (ā0 /A) ∈ U ∧ U0 p ,
t∗ (ā /A) ∈ U ∧ U1 p .
Thus we can build the following tree of formulas:
U0
U000 · · ·
U00
H U001 · · ·
@ U U010 · · ·
01
H U011 · · ·
2
That is, |= ¬(Uη0 (x̄) ∧ Uη1 (x̄)), Uη p = ∅, |= Uη0 → Uη (x̄),
|= Uη1 (x̄) → Uη (x̄), , η ∈ 2ω .
Hence we can find 2ℵ0 types in Sp∗ (A).
Corollary
Either |Sp∗ (A)| < ℵ0 or |Sp∗ (A)| ≥ 2ℵ0 .
An Open Map Theorem
Proposition 2 Let B ⊂ A, p ∈ Sn (B), q ∈ Sn (A), p ⊂ q. Then the map η : Sq∗ (A) −→
defined by η(t∗ (ā/A)) = t∗ (ā/B) is continuous.
Proof Let U (x̄)p be any basic open set in Sp∗ (B). Since L∗B ⊂ L∗A , U (x̄)q is a basic
open set in Sq∗ (A). Furthermore, η −1 (U p ) = U q . So η −1 (U p ) is open.
Now let w(x̄ ȳ) ∈ S(A), p(x̄) = w(x̄ ȳ)|x̄ ∈ S(A).
∗
∗
(A) −→ Sp∗ (A), η(q) = { φ(x̄) ∈ L∗A : φ(x̄) ∈ q}, where q = q(x̄ ȳ) ∈ Sw
(A).
Define η : Sw
Theorem 2 Let T be a complete theory in a countable language. Let w(x̄ ȳ) ∈ S(A), p(x̄)
= w(x̄ ȳ)|x̄ ∈ S(A). Let w(x̄, ȳ) be ℵ1 -isolated (that is, there is a countable subset B ⊂ A such
∗
(A) −→ Sp∗ (A) is open.
that w(x̄ ȳ)|B w(x̄, ȳ)). Then η : Sw
Proof To prove that η is open, it suffices to prove that for any Ui (x̄ȳ)w = ∅, η(Ui w ) is
open in Sp∗ (A).
Choose E(x̄ȳ, x̄ ȳ ) ∈ F E(A) with one of its equivalence classes corresponding to Ui . Denote
by A0 the set obtained by adding the parameters in E(x̄y, x y¯ ) to B. Then we still have
w(x̄, ȳ)|A0 w(x̄, ȳ). Let w0 (x̄, ȳ) = w(x̄, ȳ)|A0 , p0 (x̄) = p(x̄)|A0 .
Since L and A0 are countable, we can assume that w0 (x̄, ȳ) = {θn : θn+1 θn , n < ω}.
(1) First prove the following:
There exists an N < ω such that for all n ≥ N, ∀ȳ(θN (x̄ȳ) → ∃ȳ (θn (x̄ȳ ) ∧ E(x̄ȳ, x̄ȳ ))) ∈
p0 . Let b̄ satisfy p0 . Denote Fn = {Uj : MA∗ |= ∃ȳ(θn (b̄ȳ) ∧ Uj (b̄ȳ)), UjM is an E-class}.
Obviously, Fn ⊂ Fm , (n ≥ m). There exists a least N such that FN is minimal. Then for
all n ≥ N, Fn = FN and M |= ∀ȳ( θN (b̄ȳ) → ∃ȳ ( θn (b̄ȳ ) ∧ E(b̄ȳ, b̄ȳ ))). So the claim above is
valid.
Since M is saturated, it follows that
Sp∗ (B)
M |= ∀ȳ(θN (b̄ȳ) → ∃ȳ (w0 (b̄ȳ ) ∧ E(b̄ȳ, b̄ȳ )))
(∗∗)
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for all realizations b̄ of p0 .
(2) Let E ∗ (x̄, x̄ ) = ∀ȳ(θN (x̄ȳ) → ∃ȳ (θN (x̄ ȳ )∧E(x̄ȳ, x̄ ȳ ))) ∧∀ȳ (θN (x̄ ȳ ) → ∃ȳ(θN (x̄ȳ)∧
E(x̄ȳ, x̄ ȳ ))).
Then E ∗ (x̄, x̄ ) is an A0 -definable equivalence relation. And E ∗ (x̄, x̄ ) ∈ F E(A).
(3) Now prove that η(Ui w ) is open.
Let r(x̄) ∈ η(Ui w ). Then r(x̄) = η(q) for some q = t∗ (b̄c̄ /A) ∈ Ui w . Hence r = t∗ (b̄/A)
and Ui (b̄c̄). Let Dk be the E ∗ -class of b̄ and let Uk correspond to Dk . Then r(x̄) ∈ Uk (x̄)p .
Now for any t∗ (b̄ /A) ∈ Uk (x̄)p , we can find d¯ such that b̄ d¯ realizes w(x̄ȳ). Thus p0 =
t(b̄ /A0 ) and E ∗ (b̄, b̄ ).
From E ∗ (b̄, b¯ ) and θN (b̄c̄) we get c̄ such that θN (b̄ c̄ ) ∧ E(b̄c̄, b̄ c̄ ). By θN (b̄ c̄ ) and (∗∗)
there is c̄0 such that w0 (b̄ c̄0 ) ∧ E(b̄ c̄ , b̄ c̄0 ). Hence w(b̄ c̄0 ) and Ui (b̄ c̄0 ) = Ui (b̄ c̄ ) = Ui (b̄c̄).
Let q0 = t∗ (b̄ c̄0 /A) ∈ Ui w . Then η(q0 ) = t∗ (b̄ /A). So η is open.
Remark For a given theorem T (even if T is ω-stable ) we can not say that any type
over some subset of a model for T is ℵ1 -isolated.
Example Let T be the theory of an infinite set (with no relations other than equality).
Then T is ω-stable. Let A be a set, p(x) ∈ S1 (A). Then either p(x) contains x = a for some
a ∈ A, or p(x) = {x = a : a ∈ A}. Now let A be uncountable. Then A |= T. Choose a ∈ A. It
is easy to see that t(a/(A \ {a})) is not ℵ1 -isolated.
The following is in some sense not an open map theorem, but it is interesting.
Theorem 3 Let T be a complete theory in some language (possibly uncountable). Let
∗
w(x̄, ȳ), p(x̄) be given as in Theorem 2. Then for any nonempty Ui w ⊆ Sw
(A), any q =
∗
∗
t (b̄c̄/A) ∈ Ui w there is a Uk p such that η(q) ∈ Uk p and for any t (b̄0 /A) ∈ Uk p , any
A-isomorphism f of M with f (b̄) = b̄0 we have t∗ (f n (b̄)/A) ∈ η(Ui w ) ∩ Uk p for some
n ≥ 1.
Proof Let Ui correspond to E ∈ F E(A). Denote
E ∗ (x̄, x̄ ) = ∀ȳ∃ȳ E(x̄ȳ, x̄ ȳ ) ∧ ∀ȳ ∃ȳE(x̄ȳ, x̄ ȳ ).
One easily checks that E ∗ ∈ F E(A). Let UkM be the E ∗ -class of b̄. Then η(q) ∈ Uk p .
For any t∗ (b̄0 /A) ∈ Uk p we have Uk (b̄0 ) and E ∗ (b̄0 , b̄). Let f be an A-isomorphism of M
with f (b̄) = b̄0 . Thus E ∗ (b̄, f n (b̄)), n ≥ 1.
Let b̄c̄0 , b̄c̄1 , ..., b̄c̄m−1 , ām c̄m , ..., ān c̄n , (āi = b̄, c̄0 = c̄) be the representatives of all equivalence classes of E(x̄ȳ, x̄ ȳ ), and assume m is maximal in all such representations.
From E ∗ (b̄, f (b̄)) there are d¯0 , ..., d¯m−1 such that E(b̄c̄i , f (b̄)d¯i ) and that for any d¯ there
¯ Particularly, for f (c̄i ) there is a σ(i) ∈ {0, 1, ..., m − 1} such that
is a c̄i with E(b̄c̄i , f (b̄)d).
E(b̄c̄σ(i) , f (b̄)f (c̄i )).
It is easily seen that σ is a permutation of { 0, 1, ..., m−1}. So σ m = 1. Since E(f m (b̄)f m (c̄i ),
b̄c̄σm (i) ), E(f m (b̄)f m (c̄i ), b̄c̄i ). Hence E(f m (b̄)f m (c̄0 ), b̄c̄0 ).
It shows that Ui (f m (b̄)f m (c̄0 )). From E ∗ (b̄, f m (b̄)) and Uk (b̄) we get Uk (f m (b̄)). Now
w(x̄ȳ) = t(b̄c̄0 /A) = t(f m (b̄)f m (c̄0 )/A), t∗ (f m (b̄)f m (c̄0 )/A) ∈ Ui w and t∗ (f m (b̄)/A) ∈
Uk p .
Finally, we give several propositions dealing with the relations between the models of TA
and the models of TA∗ . For brevity, we assume L is any language. Let T ∗ denote T∅∗ .
Proposition 3 Suppose M, N |= T. Then M ≡ N iff M ∗ ≡ N ∗ .
Proof Use Shelah-Keisler’s isomorphism theorem about elementary equivalence between
models and the expansion theorem about reduced products.
Shen Yunfu
The Space of Strong Types and an Open Map Theorem
63
Proposition 4 Let M |= T. Then for any L∗ -formula φ(x̄) there exists an L-formula
φ (x̄ȳ) and b̄ ∈ M such that for any ā ∈ M , M ∗ |= φ(ā) ⇐⇒ M |= φ∗ (āb̄).
Proof We only need to note that for any E(ȳ, z̄) ∈ F E(∅), if U (ȳ) is a new relation symbol
corresponding to E(ȳ, z̄), then there is a c̄ ∈ M such that M ∗ |= U (ȳ) ↔ E(ȳ, c̄). Thus we can
replace U (ȳ) in φ(x̄) by E(ȳ, c̄) and obtain an LM -formula φ∗ (x̄b̄).
Proposition 5 N ≺ M iff N ∗ ≺ M ∗ .
Proof Use Propositoin 4.
∗
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