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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 217–221

Transitivity of hereditarily metacompact spaces

Hans-Peter A. Künzi
∗

Dedicated to Professor S. Naimpally on the occasion of his 70th birthday.

Abstract. We prove that each regular hereditarily metacompact
(monotonic) β-space has the property that the third power of any neigh-
bornet belongs to its point-finite quasi-uniformity.

2000 AMS Classification: 54E15, 54D20, 54E18, 54E25.
Keywords: point-finite quasi-uniformity, transitive quasi-uniformity, transi-
tive topological space, β-space, monotonic β-space, hereditarily metacompact,
Choquet game.

1. Introduction.

Junnila [6, Corollary 4.13] showed (see also [4, Theorem 6.21]) that in a
semistratifiable metacompact space the third power of each neighbornet belongs
to the point-finite quasi-uniformity. Similarly, in [7] it was proved that each
regular hereditarily metacompact compact space possesses the latter property.

Junnila’s result and the techniques used in [7] suggested that it should be
possible to generalize the latter result beyond (local) compactness using meth-
ods known from the theory of monotonic properties (compare [2]).

In this note we verify this conjecture by presenting a proof which shows
that each regular hereditarily metacompact (monotonic) β-space satisfies the
condition that the third power of any neighbornet belongs to its point-finite
quasi-uniformity.

Recall that a topological space is called transitive (see e.g. [4]) provided
that its finest compatible quasi-uniformity has a base consisting of transitive
entourages. Hence in particular our result implies that each regular hereditarily
metacompact (monotonic) β-space is transitive.

For basic facts about quasi-uniformities we refer the reader to [4].

∗The author acknowledges support by the Swiss National Science Foundation (under grant
20-63402.00) during stays at theUniversity of Berne, Switzerland.



218 H.-P.A. Künzi

2. Main result.

Let us first mention some pertinent definitions and recall a few well-known
facts. A regular topological space X is said to be a monotonic β-space [1]
if, for each point x ∈ X, there exists a decreasing sequence 〈Bn(x)〉n∈ω of
open neighborhood bases of X at the point x such that if Bn ∈ Bn(xn) and
Bn+1 ⊆ Bn whenever n ∈ ω and if

⋂
n∈ω Bn is nonempty, then the sequence

〈xn〉n∈ω has a cluster point. The family {〈Bn(x)〉n∈ω : x ∈ X} is called a
monotonic β-system of X.

The following results are known to hold (in the class of regular (T1-)spaces):
Each β-space is a monotonic β-space. Every monotonic p-space is a monotonic
β-space [1, Proposition 1.7]. Furthermore, every submetacompact monotonic
p-space is a p-space [2, Theorem 2.8(b)]. Recall also that each submetacompact
space is a p-space if and only if it is a w∆-space [5, Theorem 3.19].

We shall find it convenient to work with the following class of (regular)
topological spaces X that is defined in terms of a game G(X) in X, which is a
modification of certain games introduced in [3] and was suggested to us by Prof.
J. Chaber. The game G(X) is similar to the strong game of Choquet. Player I
starts the game by choosing a nonempty open set V0 and a point x0 ∈ V0. After
player I has chosen his nonempty open set Vn and xn ∈ Vn in his nth move
where n ∈ ω, player II replies with an open set Wn ⊆ Vn containing xn and
player I, in the next move, has to pick Vn+1 inside Wn. Player II wins if either
∩n∈ωWn = ∅ or the sequence 〈xn〉n∈ω has a cluster point in X. A winning
strategy for player II is a function s into the topology of X defined on all finite
sequences of moves of player I so that player II always wins when using the
function s to determine his next move.

It is readily seen that for each (regular) monotonic β-space X, player II has
a winning strategy for the game G(X). Indeed in his nth move he will choose
as Wn some (fixed) member of Bn(xn) contained in Vn.

A scattered partition (see e.g. [8, Definition 2.4]) of a topological space
X is a cover {Lα : α < γ} of X by pairwise disjoint sets such that the set
Sβ =

⋃
{Lα : α < β} is open for each β ≤ γ.

A binary relation N on a topological space X is called a neighbornet of X
if N(x) = {y ∈ X : (x,y) ∈ N} is a neighborhood at x whenever x ∈ X.
For any interior-preserving open cover C of a topological space X, we define
the neighbornet DC of X by setting DC(x) =

⋂
{C ∈ C : x ∈ C} whenever

x ∈ X. We recall that the filter on X ×X generated by the subbase {DC : C is
a point-finite open cover of X} is called the point-finite quasi-uniformity of X
(see e.g. [4]).

Lemma 2.1. Suppose that X is a hereditarily metacompact space and let O be
a neighbornet of X such that O(x) is open whenever x ∈ X. Then there is a
point-finite open cover G(X) of X such that for each member H ∈G(X) there
is xH ∈ X such that xH ∈ H ⊆ O(xH).
Proof. Choose inductively a possibly transfinite sequence 〈xα〉α of points in X
such that xα ∈ X\

⋃
β<α O(xβ) as long as possible, say whenever α < γ. Then



Transitivity 219

{O(xα) \
⋃
β<α O(xβ) : α < γ} is a scattered partition of X. According to [8,

Theorem 6.3] a topological space X is hereditarily metacompact if and only if
every scattered partition of X has a point-finite open expansion.

Hence there is a point-finite open collection {Pα : α < γ} of X such that
[O(xα) \

⋃
β<α O(xβ)] ⊆ Pα whenever α < γ. It follows that

⋃
{Pα ∩ O(xα) :

α < γ} = X and xα ∈ Pα∩O(xα) ⊆ O(xα) whenever α < γ. Therefore we can
set G(X) = {Pα ∩O(xα) : α < γ}. �

Theorem 2.2. Let O be a neighbornet of a regular hereditarily metacompact
space X. If player II has a winning strategy in the game G(X) described above,
then there exists a point-finite open family U of X such that DU ⊆ O3.

Proof. Without loss of generality we suppose that O(x) is open whenever x ∈
X. Inductively for each n ∈ ω we shall define a point-finite open family Un of
X.

By Lemma 2.1 there is a point-finite open cover U0 of X which has the
property that for each U0 ∈ U0 there is some point pU0 ∈ X such that pU0 ∈
U0 ⊆ O(pU0 ).

Let n ∈ ω. Suppose that we have defined the point-finite open family Uk+1 as
the union of families Gk+1(U) where U runs through a subfamily of Uk whenever
k < n. (In the following we distinguish between members in different families
Gk+1(U) or in U0 that denote the same set; in this way each member arises on
a well-defined level of the construction and each member V belonging to the
level Uk+1 has a unique element U in the level Uk preceding it in the sense that
V ∈Gk+1(U).)

Furthermore suppose that each Un where Uk+1 ∈Gk+1(Uk) whenever k < n
determines the sequence (U0 \ O−1(pU0 ),pU1,W0,U1 \ O−1(pU1 ),pU2,W1, . . . ,
Un−1 \ O−1(pUn−1 ),pUn,Wn−1) which describes the moves k < n of a well-
defined instance of the game G(X) in the sense that

(1) player I has used Uk \ O−1(pUk) and some well-defined point pUk+1 of
Uk \O−1(pUk) in his k

th-move whenever k < n,
and (2) player II has chosen the set Wk according to his winning strategy in

his kth move whenever k < n.
In particular note that each Wk is determined by the preceding moves of

player I and that each Uk \O−1(pUk) 6= ∅ whenever k < n. Call a member Un
of Un suitable (in Un) if Un 6⊆ O−1(pUn).

Assume now that Un is a suitable member of Un. Suppose that player I
continues the beginning of the game G(X) associated with Un by choosing
Un\O−1(pUn) and any x ∈ Un\O−1(pUn) in his nth move. Then player II finds
Wn(. . . ,Un,x) according to his winning strategy such that x ∈ Wn(. . . ,Un,x) ⊆
Un \O−1(pUn).

By hereditary metacompactness and regularity of X there exists a point-
finite open cover Vn+1(Un) of Un\O−1(pUn) such that the closures of its mem-
bers are all contained in Un \O−1(pUn). Consider the neighbornet of the sub-
space Un \O−1(pUn) of X determined by the neighborhoods Wn(. . . ,Un,x) ∩



220 H.-P.A. Künzi⋂
{E ∈ Vn+1(Un) : x ∈ E}∩ O(x) whenever x ∈ Un \ O−1(pUn). By Lemma

2.1 there exists a point-finite open cover Gn+1(Un) of Un \O−1(pUn) such that
for each Un+1 ∈ Gn+1(Un) there is some point pUn+1 ∈ Un \ O−1(pUn) sat-
isfying pUn+1 ∈ Un+1 ⊆ Wn(. . . ,Un,pUn+1 ) ∩

⋂
{E ∈ Vn+1(Un) : pUn+1 ∈

E}∩O(pUn+1 ).
Set Un+1 =

⋃
{Gn+1(Un) : Un is a suitable member of Un}. Note that Un+1

is a point-finite open family of X. Observe also that for each suitable Un of
Un the closures of all members of Gn+1(Un) are contained in Un \ O−1(pUn)
because Vn+1(Un) had the latter property.

Furthermore by the construction above it is readily checked that for each
member Un+1 ∈ Gn+1(Un) we have constructed the moves k < n + 1 of the
instance of the game G(X) associated with Un+1 by adding to the (unique)
sequence of moves associated with Un the nth moves (Un\O−1(pUn),pUn+1,Wn)
of player I and player II, respectively, where Wn = Wn(. . . ,Un,pUn+1 ).

Claim 2.3. There exists a point-finite family U of open sets of X such that
the family H = {U ∩O−1(pU ) : U ∈U} covers X.

We shall show that our claim holds for the family U =
⋃
n∈ω Un:

Suppose that for some x ∈ X there are infinitely many sets in U containing
x. Consider the family S of all sets in U containing x. Since each family Un
is point-finite, we conclude by König’s Lemma [9] and the definition of the
families Un+1 that in S there exists a sequence 〈Un〉n∈ω such that for each
n ∈ ω, Un+1 ∈ Gn+1(Un). We shall show next that such a sequence does not
exist.

Note first that the sequence 〈Un \O−1(pUn),pUn+1〉n∈ω yields the moves of
player I in an instance of the game G(X) where player II uses his winning
strategy to find the sets Wn = Wn(. . . ,Un,pUn+1 ) whenever n ∈ ω.

By the construction of the family Gn+1(Un), Un+1 ⊆ Un+1 ⊆ Un\O−1(pUn)
and pUn+1 ∈ Un+1 ⊆ Wn(. . . ,Un,pUn+1 ) ⊆ Un \O−1(pUn) whenever n ∈ ω.

Since x ∈ ∩n∈ωUn and thus x ∈ ∩n∈ωWn(. . . ,Un,pUn+1 ), we conclude that
〈pUn〉n∈ω has a cluster point z in X. Thus pUn ∈ O(z) for infinitely many
n ∈ ω. But also z ∈ Un+1 whenever n ∈ ω, because for each n ∈ ω a tail
of the sequence 〈pUn〉n∈ω is contained in Un+1. Since Un+1 ∩ O−1(pUn) = ∅
whenever n ∈ ω, we see that z 6∈ O−1(pUn) whenever n ∈ ω —a contradiction.
We conclude that the family U is point-finite.

Suppose that some point x ∈ X is not contained in any set U ∩ O−1(pU )
where U ∈ U. Since U0 is a cover of X, there exists U0 ∈ U0 such that
x ∈ U0. Suppose that n ∈ ω and sets Uk (k ≤ n) have inductively been
defined such that x ∈ Uk+1 ∈ Gk+1(Uk) (k < n). By our assumption, we have
that x ∈ Un \ O−1(pUn). In particular, Un is suitable in Un. Since Gn+1(Un)
covers Un\O−1(pUn), there exists Un+1 ∈Gn+1(Un) such that x ∈ Un+1. This
concludes the induction. Of course, x ∈ ∩n∈ωUn. But as we just noted above
such a sequence 〈Un〉n∈ω cannot exist. Hence H is a cover of X.



Transitivity 221

Finally we show that DU ⊆ O3. Let x ∈ X. By the claim verified above
there exists U ∈ U such that x ∈ U ∩ O−1(pU ). Furthermore, we see that
DU(x) =

⋂
{V ∈ U : x ∈ V} ⊆ U ⊆ O(pU ) by the selection of the sets U

belonging to U. Since we have that x ∈ O−1(pU ), there exists a point y ∈
O(x) ∩ O−1(pU ). We now conclude that y ∈ O(x) and pU ∈ O(y). It follows
that pU ∈ O2(x) and, furthermore, that O(pU ) ⊆ O3(x). As a consequence, we
see that DU(x) ⊆ O(pU ) ⊆ O3(x), which confirms the assertion. �

Acknowledgements. We would like to thank Prof. J. Chaber for sugges-
tions that led to several improvements of this note.

References

[1] J. Chaber, On point-countable collections and monotonic properties, Fund. Math. 94
(1977), 209–219.

[2] J. Chaber, M.M. Čoban and K. Nagami, On monotonic generalizations of Moore spaces,

Čech complete spaces and p-spaces, Fund. Math. 84 (1974), 107–119.
[3] J. Chaber and R. Pol, On hereditarily Baire spaces and σ-fragmentability of mappings

and Namioka property, preprint.
[4] P. Fletcher and W.F. Lindgren, Quasi-uniform Spaces, Lecture Notes Pure Appl. Math.

77, Dekker, New York, 1982.
[5] G. Gruenhage, Generalized metric spaces, in: Handbook of Set-theoretic Topology, K.

Kunen and J.E. Vaughan, eds., North-Holland Amsterdam, 1984, pp. 423–501.
[6] H.J.K. Junnila, Neighbornets, Pacific J. Math. 76 (1978), 83–108.
[7] H.J.K. Junnila, H.P.A. Künzi and S. Watson, On a class of hereditarily paracompact

spaces, Topology Proc. 25 Summer (2000), 271–289; Russian translation in: Fundam.
Prikl. Mat. 4 (1998), 141–154.

[8] H.J.K. Junnila, J.C. Smith and R. Telgársky, Closure-preserving covers by small sets,
Topology Appl. 23 (1986), 237–262.

[9] D. König, Sur les correspondances multivoques des ensembles, Fund. Math. 8 (1926),

114–134.

Received September 2001

Revised July 2002

Hans-Peter A. Künzi

Dept. Math. and Appl. Math., University of Cape Town, Rondebosch 7701,
South Africa

E-mail address : kunzi@maths.uct.ac.za