CARDINALITY RESTRICTIONS ON POWER
HOMOGENEOUS T5 COMPACTA
G. J. RIDDERBOS
Abstract. We show that the cardinality of power homogeneous
T5 compacta X is bounded by 2c(X) . This answers a question of
J. van Mill, who proved this bound for homogeneous T5 compacta.
We further extend some results of I. Juhász, P. Nyikos and Z. Szentmiklóssy and as a corollary we prove that consistently every power
homogeneous T5 compactum is first countable. This improves a
theorem of R. de la Vega who proved this consistency result for
homogeneous T5 compacta.
1. Introduction
All spaces are assumed to be Hausdorff and all cardinal functions are
assumed to be infinite. By c(X), πχ(X), t(X) and χ(X) we denote the
cellularity, π-character, tightness and character. By πχ(p, X), t(p, X)
and χ(p, X) we denote the local versions of the π-character, tightness
and character at the point p ∈ X. A space X is called homogeneous
if for every x, y ∈ X, there is a homeomorphism h of X such that
h(x) = y. A space X is called power homogeneous if X µ is homogeneous
for some cardinal number µ.
Jan van Mill has shown in [8, Corollary 2.6] that if X is a power
homogeneous compactum, then |X| ≤ 2πχ(X)c(X) . By deep results
of Šapirovskiı̆, every T5 compactum contains points of countable πcharacter. So if a T5 compactum X is in addition also homogeneous,
then the π-character of X is countable and therefore the cardinality
of X is bounded by 2c(X) . This was noted by van Mill in [8] and he
asked whether the same inequality holds for power homogeneous T5
compacta. A partial answer was provided by A. Bella in [4]. Below
we provide a positive answer to van Mill’s question by showing that
if X is a power homogeneous T5 compactum, then its π-character is
countable.
Date: September 13, 2006.
2000 Mathematics Subject Classification. 03E35, 54A25, 54B10, 54D15, 54D30.
Key words and phrases. Power homogeneity, compactum, T5 , tightness,
cellularity.
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G. J. RIDDERBOS
Partially answering another question of J. van Mill, it was shown
by I. Juhász, P. Nyikos and Z. Szentmiklóssy in [6] that consistently
every homogeneous T5 compactum is of cardinality ≤ c. In [15], R. de
la Vega proved that every homogeneous compact space of countable
tightness is of cardinality ≤ c. Applying the results in [6] and a result
of F. D. Tall from [14], it follows from the bound proved by de la Vega
that consistently every homogeneous T5 compactum is first countable.
The bound proved by de la Vega for homogeneous compacta was also
proved for power homogeneous compacta by A. V. Arhangel0 skiı̆, J.
van Mill and G. J. Ridderbos in [2]. It is therefore natural to ask
whether the consistency result about first countability of homogeneous
T5 compacta may be extended to the class of power homogeneous T5
compacta. This was brought to my attention by F. D. Tall at the 2006
Prague Topological Symposium. Below we answer this question by
extending some results from [6] to power homogeneous T5 compacta.
This partially solves [3, Problem 3.17].
Not every power homogeneous T5 compactum is homogeneous as the
closed unit interval demonstrates.
2. power homogeneous T5 compacta
As in [6], a compactum is an infinite compact space. By ∆(X, µ)
we denote the diagonal in the space X µ , which consists of all points
x ∈ X µ such that xα = xβ for all α, β ∈ µ. The space X µ is
called ∆-homogeneous if any two points on the diagonal in X µ can
be mapped onto each other by a homeomorphism of X µ . A space X
is called ∆-power homogeneous if for some cardinal µ, the space X µ is
∆-homogeneous. This notion was introduced in [10] and it has been
shown by the author in [9] that a space is ∆-power homogeneous if and
only if it is power homogeneous.
The following statement is a consequence of known results and it
improves Corollaries 2.8 & 2.9 from [8].
Proposition 2.1. Suppose X is a power homogeneous compact space.
Then 2χ(X) ≤ 2πχ(X)c(X) and under GCH, χ(X) ≤ πχ(X)c(X). Furthermore, one of the following two statements is true:
(1) πχ(X) = χ(X),
(2) πχ(X) < χ(X) and χ(x, X) = χ(y, X) for all x, y ∈ X.
Proof. We first prove that either (1) or (2) is true. Let κ = min{χ(x, X) :
x ∈ X}. It follows from [2, Corollary 2.5] that χ(X) ≤ κ · πχ(X). So
if κ ≤ πχ(X), then (1) is true and if πχ(X) < κ, then χ(x, X) = κ for
all x ∈ X.
CARDINALITY RESTRICTIONS
3
The inequality 2χ(X) ≤ 2πχ(X)c(X) only needs proof if (2) is true. In
this case X has no isolated points and it follows from the Čech-Pospišil
Theorem that 2χ(X) ≤ |X|. The inequality |X| ≤ 2πχ(X)c(X) was proved
by van Mill in [8] for power homogeneous compacta and it also holds
for power homogeneous regular spaces by [10, Corollary 3.5].
We will use notation from [10]. So if µ is an infinite cardinal and
A ⊆ µ, then by πA we denote the natural projection of X µ onto X A .
By π we denote π{0} , the projection onto the first co-ordinate and if
B ⊆ A, then by πA→B we denote the projection of X A onto X B . If
p ∈ ∆(X, µ) and U is a local π-base at π(p) in X, then the collection
U(A) is given by
n
hY
i
o
−1
πA→B
Ub : B ∈ [A]<ω , ∀b ∈ B Ub ∈ U .
b∈B
Note that U(A) is a local π-base at pA in X A and |U(A)| ≤ |A| · |U|.
Lemma 2.2. Suppose X is power homogeneous and let κ be some
infinite cardinal. If the set D = {x ∈ X : πχ(x, X) ≤ κ} is dense in
X, then πχ(X) ≤ κ.
Proof. It suffices to show that for every x ∈ X there is some E ∈ [D]≤κ
such that x ∈ E.
Let µ be large enough so that X µ is homogeneous. Fix p ∈ ∆(D, µ)
and let x ∈ ∆(X, µ) be arbitrary. Since π(p) ∈ D, we may fix a local
π-base U at π(p) in X with |U| ≤ κ. By homogeneity, let h be a
homeomorphism of X µ with h(p) = x.
Applying [2, Theorem 2.2] and using the fact that D is dense in X, we
recursively construct a collection of co-ordinates A ∈ [µ]≤κ satisfying
the following condition: For every U ∈ U(A), there is some e ∈ πA−1 [U ]
such that
(1) πh(e) ∈ D,
(2) hπA−1 (eA ) ⊆ π −1 (πh(e)).
The construction of A is identical to the construction of a similar set
in the proof of [10, Theorem 3.1].
For every U ∈ U(A) we pick an element e satisfying the above conditions. We let the set E consist of all elements of the form πh(e) obtained
in this way. By construction we have E ⊆ D and since |A| ≤ κ, we
also have that |E| ≤ κ. It remains to show that π(x) ∈ E.
So let V be an arbitrary open neighbourhood of π(x) in X. Then
p ∈ h−1 π −1 [V ] and therefore the set W which is given by πA h−1 π −1 [V ]
is an open neighbourhood of pA in X A . Since U(A) is a local π-base
at pA , we may find U ∈ U(A) such that U ⊆ W . By construction,
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G. J. RIDDERBOS
there is an element e ∈ πA−1 [U ] satisfying the above conditions such
that πh(e) ∈ E.
Since eA ∈ U ⊆ W = πA h−1 π −1 [V ], it follows that
πA−1 (eA ) ∩ h−1 π −1 [V ] 6= ∅.
and therefore
hπA−1 (eA ) ∩ π −1 [V ] 6= ∅.
But hπA−1 (eA ) ⊆ π −1 (πh(e)), so it follows that πh(e) ∈ V and therefore
E ∩ V 6= ∅. Since V was an arbitrary open neighbourhood of π(x), we
have shown that π(x) ∈ E and this completes the proof.
It was proved by Šapirovskiı̆ that if a compact space X does not
map continuously onto Iω1 , then the set {x ∈ X : πχ(x, X) ≤ ω} is
dense in X, see [5, 3.18 & 3.20]. Since compact T5 spaces do not map
continuously onto Iω1 , we obtain the following corollary;
Corollary 2.3. Suppose X is a power homogeneous T5 compactum.
Then πχ(X) ≤ ω.
The following generalizes Theorem 3.2 from [8].
Theorem 2.4. Suppose X is a power homogeneous T5 compactum.
Then |X| ≤ 2c(X) .
Proof. This follows from Corollary 2.3 and the fact that |X| ≤ 2πχ(X)c(X)
(see [8, Corollary 2.6]) .
Theorem 2.5. Suppose X is a power homogeneous T5 compactum.
Then for all x, y ∈ X, χ(x, X) = χ(y, X).
Proof. Since πχ(X) ≤ ω by Corollary 2.3, the statement follows from
Proposition 2.1.
The following result generalizes [4, Corollary 3]. Note that if a power
homogeneous T5 compactum X contains isolated points, then by the
previous result X is first-countable and therefore χ(X) ≤ c(X).
Corollary 2.6. If X is a power homogeneous T5 compactum without
isolated points, then |X| = 2χ(X) and under GCH, χ(X) ≤ c(X).
Proof. By Arhangel0 skiı̆’s Theorem we have |X| ≤ 2χ(X) . It follows
from the Čech-Pospišil Theorem and Theorem 2.5 that 2χ(X) ≤ |X|. The following result generalizes [4, Corollary 4]. The proof is almost
identical to the proof of [8, Corollary 3.5].
Corollary 2.7 (GCH). Let X be a power homogeneous compactum and
let κ = c(X). If X does not contain a copy of βκ, then χ(X) ≤ c(X).
CARDINALITY RESTRICTIONS
5
Proof. Apply the same reasoning as in the proof of [8, Corollary 3.5]
to conclude that X has a dense set of points of local π-character ≤ κ.
Now apply Lemma 2.2 to conclude that πχ(X) ≤ κ and continue as in
[8].
3. Consistency results
Lemma 3.1. Suppose X is a power homogeneous compactum and
t(p, X) ≤ κ for some p ∈ X. If πχ(x, X) ≤ κ, then t(x, X) ≤ κ.
Proof. Let µ be large enough so that X µ is homogeneous. We will
abuse notation and assume that p, x ∈ ∆(X, µ) and t(π(p), X) ≤ κ and
πχ(π(x), X) ≤ κ. Since ∆(X, µ) is homeomorphic to X, it suffices to
show that if Y ⊆ ∆(X, µ) and x ∈ Y , then x ∈ Z for some Z ∈ [Y ]≤κ .
Fix a homeomorphism h of X µ such that h(p) = x. By [2, Theorem 2.2] and the fact that πχ(π(x), X) ≤ κ, we may find a set of
co-ordinates A ⊆ µ such that |A| ≤ κ and
hπA−1 (pA ) ⊆ π −1 (π(x)).
Since X is compact and |A| ≤ κ, it follows that t(pA , X A ) ≤ κ (see for
example [5, 5.9]). But πA is continuous and therefore pA is contained
in the closure of πA h−1 (Y ) in X A . Therefore we may find a set W ⊆
h−1 (Y ) with |W | ≤ κ and pA ∈ πA [W ]. Let Z = h[W ] ⊆ Y . By
compactness, it follows that
πA−1 (pA ) ∩ W 6= ∅.
and therefore
hπA−1 (pA ) ∩ Z 6= ∅.
Since hπA−1 (pA ) ∩ ∆(X, µ) = {x} and Z ⊆ Y ⊆ ∆(X, µ) it must be
the case that x ∈ Z. Since |Z| = |W | ≤ κ and Z ⊆ Y this completes
the proof.
Corollary 3.2. Suppose X is a power homogeneous compactum. Then
one of the following statements is true:
(1) πχ(X) = t(X),
(2) πχ(X) < t(X) and t(x, X) = t(y, X) for all x, y ∈ X.
Proof. The proof is an application of Lemma 3.1 and it is similar to the
proof of Proposition 2.1, note in addition for (1) that πχ(X) ≤ t(X)
for any compact space X. This was proved by Šapirovskiı̆ in [12]. Corollary 3.3. If X is a power homogeneous T5 compactum, then for
all x, y ∈ X, t(x, X) = t(y, X).
Proof. This follows from Corollary 2.3 and Corollary 3.2.
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G. J. RIDDERBOS
The following notion was used in [6], the terminology we introduce
here is mentioned before Problem 3.12 in [6].
Definition 3.4. Suppose X is a topological space. We call a point
y ∈ X a pseudo P-point if every Gδ -set containing y has non-empty
interior.
A set D ⊆ X is called Gδ -dense, if D intersects every non-empty
Gδ -subset of X. Note that if the set of all pseudo P -points of X is
Gδ -dense in X, then every point of X is a pseudo P -point.
The following lemma is proved in [6]. Since we do not allow cardinal
functions to be finite, we exclude the possibility of isolated points in
the statement of the following result.
Lemma 3.5 ([6, Lemma 2.1]). Let Y be a locally compact space without
isolated points. The set of points y which fail to satisfy at least one of
the following conditions is dense in Y :
(1) πχ(y, Y ) ≤ ω,
(2) y is a pseudo P-point.
In particular, not all points of Y can satisfy both (1) and (2).
The following theorem is a slight modification of Theorem 2.7 in [6].
A space satisfies property wD(κ) if every closed discrete subspace D of
cardinality κ has a subset D0 of cardinality κ which can be expanded
to a discrete collection of open sets Ud such that Ud ∩ D0 = {d} for all
d ∈ D0 . This notion was introduced in [6].
Theorem 3.6. If X is a compact space satisfying wD(ℵ1 ) hereditarily
and the set of points of uncountable tightness in X is Gδ -dense, then
every point of X is a pseudo P-point.
Proof. By the remarks made after Definition 3.4, it suffices to show
that the set of pseudo P -points of X is Gδ -dense in X. It is proved
in [6, Theorem 2.7], that if some free ω1 -sequence converges to y, then
y is a pseudo P -point of X. Therefore it suffices to show that every
non-empty Gδ -subset of X contains a limit point of some converging
free ω1 -sequence.
So let G be a non-empty Gδ -subset of X. Without loss of generality
we may assume that G is closed. Then by assumption, G contains
points of uncountable tightness (in X) and therefore the tightness of
G is uncountable. This follows for example from [1, 2.2.14]. Since G is
compact we apply [7, Theorem 1.2] to conclude that G contains a limit
point of some converging free ω1 -sequence. Since G is closed in X, this
sequence is also a converging free ω1 -sequence in X and this completes
the proof.
CARDINALITY RESTRICTIONS
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Corollary 3.7. If X is a power homogeneous T5 compactum satisfying
wD(ℵ1 ) hereditarily, then X is countably tight and hence of cardinality
≤ c.
Proof. If X contains an isolated point, then it contains a point of countable tightness and therefore by Corollary 3.3 the tightness of X is
countable. If X does not contain isolated points, then it follows from
Corollary 2.3, Corollary 3.3, Lemma 3.5 and Theorem 3.6 that X is
countably tight.
In both cases it follows from [2, Corollary 3.7] that |X| ≤ c.
As in [6, Theorem 2.8], it follows from the previous result that if
at least ℵ2 Cohen reals are added to any model V of ZFC, the resulting extension W has the property that every power homogeneous T5
compactum is countably tight and of cardinality ≤ c.
It was shown by F. D. Tall in [14] that under the assumption 2ℵ0 <
ℵ1
2 , every locally compact normal space is weakly ℵ1 -collectionwise
Hausdorff. It follows from the remarks after Definition 1.3 in [6] that in
such a model every locally compact T5 space satisfies property wD(ℵ1 )
hereditarily. As a consequence of Corollary 2.6 and Corollary 3.7 we
obtain the following strengthening of Theorem 3.3 in [15].
Corollary 3.8 (2ℵ0 < 2ℵ1 ). Every power homogeneous T5 compactum
is first countable.
Section 3 of [6] contains some conditions under which a homogeneous
compact space is first countable. Using the techniques of this paper
one can generalize the proofs presented there also to the case of power
homogeneous compact spaces. This applies to Theorems 3.2, 3.6 and
3.11 of [6]. In particular this yields a quick proof of the fact that
every monotonically normal power homogeneous compactum is first
countable (see the proof of Theorem 3.6 in [6]). Details will appear in
[11].
Recently, F. D. Tall has constructed a model in which power homogeneous T5 compacta satisfy the conditions of Theorem 3.2 in [6].
In this model 2ℵ0 = 2ℵ1 , so it is consistent with this cardinal equality
that power homogeneous T5 compacta are first countable. This should
be compared with Corollary 3.8. Details of this model will appear in
[13]. I would like to thank Frank Tall for bringing this result to my
attention.
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Faculty of Sciences, Division of Mathematics, Vrije Universiteit,
De Boelelaan 1081a , 1081 HV Amsterdam, the Netherlands
E-mail address: [email protected]
URL: http://www.math.vu.nl/~gfridder