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Houston Journal of Mathematics
c 2012 University of Houston
Volume 38, No. 3, 2012
A NOTE ON CLOSED DISCRETE SUBSETS
OF SEPARABLE (a)-SPACES
CHARLES J.G. MORGAN AND SAMUEL G. DA SILVA
Communicated by Yasunao Hattori
Abstract. We show that the existence of a T1 separable space with an uncountable closed discrete subset which satisfies relative versions of property
(a) and local compactness implies the existence of small dominating families
in the family of functions of ω1 into ω. Considering well-known relationships between small dominating families and large cardinals, it follows that
if Y is an uncountable closed discrete subset of a T1 separable (a)-space
X then there is no way to prove within ZFC that Y satisfies relative local
compactness.
1. Introduction.
Arhangel’skii and Genedi ([2]) introduced the programme of considering relative versions of topological properties in the late 1980s. The programme has been
taken up widely since then. See, for example, the survey papers [1] and [17], the
papers referred to there, [4] and [5], and many others besides.
The second author, improving on a result of Watson ([16]), contributed to this
programme in [14]. He showed there that if there is a separable space X with an
uncountable closed and discrete subset which is relatively countably paracompact
in X then there is a dominating family in ω1 ω of size not larger than 2ω . The
latter is well known as being related to large cardinals as we explain below.
2000 Mathematics Subject Classification. Primary 54D20, 54A25, 54A35; Secondary 54B05,
54D45, 03E55.
Key words and phrases. Relative topological properties, locally compact spaces, property
(a), countable paracompactness.
The second author’s research was supported by post-doc grant CAPES Foundation, Ministry
of Education of Brazil, Proc. 5603/09-9.
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CHARLES MORGAN AND SAMUEL GOMES DA SILVA
Countable paracompactness per se is one of the important topological covering
properties. In [10], [11], [12], [13] and [14] we proved various results concerning it.
We were often also able to prove similar results for another key covering property,
Matveev’s property (a).
In this brief note we tie up a loose end from these papers by improving on
Theorem 2.4 of [12] and proving for property (a) an analogue of the relative
result from [14] mentioned above.
Definition 1. (The mod countable order.) For any uncountable set of ordinals
A let A ω be the collection of functions from A to ω. For f , g ∈ A ω write f 6∗ g
if { α ∈ A : g(α) < f (α) } is countable.
Definition 2. A subset of an ordered set of functions is said to be a dominating
family if it is cofinal in the order.
For example, D ⊆ ω1 ω is a dominating family in the mod countable order if
(∀f ∈ ω1 ω)(∃g ∈ D)[f 6∗ g]. It is well-known that cf (hω1 ω, 6∗ i) = cf (hω1 ω, 6i),
where f 6 g means f (α) 6 g(α) for every α < ω1 (see [3]).
Jech and Prikry ([6]), using Dodd and Jensen’s results on the core model,
showed that “2ω < 2ω1 ” + “2ω regular” + “there is a dominating family in
hω1 ω, 6∗ i of cardinality 2ω ” implies that “there is an inner model with a measurable cardinal”. They also showed that there can be no dominating family of size
less than 2ω1 in hω1 ω, 6∗ i if either 2ω is a real-valued measurable cardinal or if
2ω < min({2ω1 , ℵω1 }).
We continue by reviewing some familiar general topological definitions. Throughout this paper, X denotes a topological space. For x ∈ X, “Ux is an open neighbourhood of x” always means that Ux is an open subset of X and x ∈ Ux .
Definition 3. A set D ⊆ X is discrete if every x ∈ D has an open neighbourhood
Ux such that |Ux ∩ D| = 1.
Definition 4. Let { Y } ∪ Z ⊆ P(X). The star in X induced by Y and Z,
S
St(Y, Z), is {Z ∈ Z : Y ∩ Z 6= ∅ }.
Definition 5 ([7]). X satisfies property (a) (or is an (a)-space) if for every open
cover U of X and every dense set D ⊆ X there is F ⊆ D such that F is a closed
and discrete subset of X and St(F, U) = X.
Definition 6. A family A ⊆ P(X) is locally finite if every x ∈ X has an open
neighbourhood Ux such that {A ∈ A : A ∩ Ux =
6 ∅ } is finite.
The following definition is given in order to introduce relative versions of property (a).
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Definition 7. Let A, B ⊆ X. We say that A is locally finite in B if every x ∈ B
has an open neighbourhood Ux such that Ux ∩ A is finite. If A = B, then A is
locally finite in itself. If B = X, then A is locally finite.
Note that A ⊆ X is locally finite in the sense of Definition 7 if and only if
{ {x } : x ∈ A} is locally finite in that of Definition 6. We also mention that if A
is closed and locally finite in itself then A is locally finite in any set B ⊆ X.
Definition 8. Let X be a topological space and Y ⊆ X. We consider the
following relative topological properties – notice the phrase “in X” in each.
(i) ([1], [17]). Y is compact in X if every open cover U of X has a finite
S
subfamily V such that Y ⊆ V.
(ii) ([1], [17]). Y is locally compact in X if every y ∈ Y has a neighbourhood
Vy such that Vy is compact in X.
(iii) ([1], [14]). Y is (countably) paracompact in X if for every (countable) open
cover U of X there is a family V of open subsets of X such that V is locally finite
at each point of Y , V refines U (that is, for every V ∈ V there is U ∈ U such that
S
V ⊆ U ) and Y ⊆ V.
(iv) ([8]). Y has property (a) in X (or is relatively (a) in X) if for every open
cover U of X and every dense set D ⊆ X there is C ⊆ D such that C is a closed
and discrete subset of X and Y ⊆ St(C, U).
(v) Y has property (a)0 in X (or is relatively (a)0 in X) if for every open cover
U of X and every dense set D ⊆ X there is C ⊆ D such that C is a closed subset
of X which is locally finite in itself and Y ⊆ St(C, U).
(vi) Y has weak property (a) in X (or is relatively weakly (a) in X) if for every
open cover U of X and every dense set D ⊆ X there is C ⊆ D such that C has
no accumulation points in Y and Y ⊆ St(C, U).
(vii) Y has weak property (a)0 in X (or is relatively weakly (a)0 in X) if for
every open cover U of X and every dense set D ⊆ X there is C ⊆ D such that C
is locally finite in Y and Y ⊆ St(C, U).
We introduce here the notions defined in (v)−(vii). Notice that, by elementary
properties regarding accumulation points in T1 spaces, if X is T1 then (iv) ⇔ (v)
and (vi) ⇔ (vii).
The following is an easy relative version of a result from [7].
Proposition 1.1. Let Y ⊆ X. If Y is paracompact in X then Y has weak
property (a)0 in X.
Proof. Let U be an open cover of X and D a dense subset of X. By hypothesis,
there is a family of open sets V such that V refines U, V is locally finite at
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CHARLES MORGAN AND SAMUEL GOMES DA SILVA
S
each point of Y and Y ⊆ V. For every V ∈ V pick xV ∈ V ∩ D and let
C = {xV : V ∈ D}. Then C is a subset of D which is locally finite in Y and
Y ⊆ St(C, V) ⊆ St(C, U).
Lemma 1.2. If Y ⊆ X is compact in X and H ⊆ X is a closed subset of X
which is locally finite in itself, then F = H ∩ Y is a finite set.
Proof. Let h Ux : x ∈ H i be a collection of open sets witnessing that H is
locally finite in itself and consider the open cover {Ux : x ∈ H } ∪ {X \ H }. As
Y is compact in X there are x1 , . . . xn ∈ H such that Y ⊆ Ux1 ∪ · · · ∪ Uxn ∪ X \ H.
Hence F ⊆ (Ux1 ∩ H) ∪ · · · ∪ (Uxn ∩ H), a finite union of finite sets.
2. The Main Theorem
Theorem 2.1. The existence of a T1 separable space X with an uncountable
closed and discrete subset which is also locally compact in X and relatively (a) in
X implies the existence of a dominating family in hω1 ω, 6i of size not larger than
2ω .
Proof. Let X be a T1 topological space with a countable dense set and an
uncountable closed and discrete subset which is locally compact in X and has
property (a) in X. We may, without loss of generality, take these sets to be ω
and ω1 \ ω respectively. Note that, under this assumption, ω1 \ ω has no isolated
points in X.
For each β ∈ ω1 \ ω pick an open neighbourhood Uβ of β such that
(1) Uβ ∩ (ω1 \ ω) = {β} and (2) Uβ ⊆ Kβ , where Kβ is compact in X.
Let Fcd ⊆ P(ω) be the family of all closed discrete subsets of the dense set ω.
For each C ∈ Fcd define a function fC : (ω1 \ ω) → ω by
max(Uβ ∩ C) if Uβ ∩ C 6= ∅
fC (β) =
0
otherwise.
It is easy to see that the functions in F = {fC : C ∈ Fcd } are well defined (by
the Lemma of the previous section).
We claim that F = {fC : C ∈ Fcd } is a dominating family in h(ω1 \ω) ω, 6i; this
is sufficient for us. Indeed, consider an arbitrary function g : (ω1 \ ω) → ω. Let
U be the open cover of X given by
U = {X \ (ω1 \ ω)} ∪ {Uβ \ g(β) : β ∈ (ω1 \ ω)}.
Note that, by (1), for each β ∈ (ω1 \ ω), the open set Uβ \ g(β) is the only
element of U that contains β. As ω1 \ ω has property (a) in X, there is a closed
CLOSED DISCRETE SUBSETS OF SEPARABLE (a)-SPACES
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discrete subset C ⊆ ω such that ω1 \ ω ⊆ St(C, U). It is easy to see that fC
dominates g, since for every β ∈ (ω1 \ ω) we must have
(Uβ \ g(β)) ∩ C 6= ∅
and therefore fC (β) > g(β), as desired.
It is immediate that one can refine the conclusion of the theorem to “. . . of size
not larger than ddc1 (X)”, where ddc1 (X) is the size of the smallest dominating
family in hFcd , ⊆i. (See [11], [14] for the general definition of ddc1 (X).)
Note that although local finiteness is the key to the preceding proof we do
have to use the full property (a), and not merely its weaker version (a)0 ( (v) of
Definition 8), because T1 is needed in order to guarantee that U is in fact an open
covering.
Considering Jech and Prikry’s results already mentioned, we also established
the following:
Corollary 2.2. Assume 2ω is regular and 2ω < 2ω1 . Under these conditions, if
there is a T1 separable (a)-space that includes an uncountable closed and discrete
subset which is also locally compact in X then there is an inner model with a
measurable cardinal.
The following simple rephrasing of the preceding corollary shows how Topology
and Set Theory are inextricably linked when one considers our discussion.
Fact 1. Let X be a separable, T1 , (a)-space, Y ⊆ X closed and discrete and
assume that cf (2ω ) = 2ω < 2ω1 . Then, at least one of the following statements
hold:
(i) Y is countable.
(ii) Y is not locally compact in X.
(iii) There is an inner model with a measurable cardinal.
It follows that if Y is an uncountable closed discrete subset of a T1 separable
(a)-space X then there is no way to prove within ZFC that Y satisfies relative
local compactness.
3. Notes and Questions
We close by raising the question of which other (relative) properties of (uncountable) closed, discrete sets, apart from the ones used here and relative countable paracompactness, allow one to derive the existence of small dominating families in ω1 ω.
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CHARLES MORGAN AND SAMUEL GOMES DA SILVA
By previous results of the authors, topological (or any kind of) properties that
imply the existence of small dominating families cannot hold under a certain
weak parameterized diamond principle, Φ(ω, <) ([11]) (one of the family of such
principles introduced by Moore, Hrušák and Džamonja in [9]).
Acknowledgements. The authors wish to thank the anonymous referee for their
careful reading of the paper and for calling our attention to the paper of Matveev,
Pavlov and Tartir ([8]), where the relative version of property (a) we use in this
paper (see Definition 8 (iv)) was first defined.
This work was done during the postdoctoral stay of the second author at
UNAM, Morelia, Michoacán, México, and he would like to thank the members
of the Instituto de Matemáticas (especially Michael Hrušák and Salvador Garcı́aFerreira) for their hospitality.
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Received July 22, 2010
Revised version received August 31, 2010
Second revision received September 1, 2010
Department of Mathematics, University College London, Gower Street, London,
WC1E 6BT, UK, and
Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Avenida
Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal
E-mail address: [email protected]
Instituto de Matemática, Universidade Federal da Bahia, Campus de Ondina, Av.
Adhemar de Barros, S/N, Ondina, CEP 40170-110, Salvador, BA, Brazil
E-mail address: [email protected]