For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of
continuous real-valued functions on $K$ endowed with the weak (pointwise)
topology. In this paper we address the following basic question which seems to
be open: Suppose that $K$ is an infinite (metrizable) compact space. Is it true
that $C_w(K)$ and $C_p(K)$ are homeomorphic? We show that the answer is "no",
provided $K$ is an infinite compact metrizable $C$-space. In particular our
proof works for any infinite compact metrizable finite-dimemsional space $K$

Krupski, Mikołaj

English

arXiv

Mikołaj Krupski

Springer - Publisher Connector

RACSAM (2016) 110:557–563
DOI 10.1007/s13398-015-0248-0
ORIGINAL PAPER
On the weak and pointwise topologies in function spaces
Mikołaj Krupski1
Received: 10 April 2015 / Accepted: 31 August 2015 / Published online: 15 September 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract For a compact space K we denote by Cw(K ) (Cp(K )) the space of continuous
real-valued functions on K endowed with the weak (pointwise) topology. In this paper we
address the following basic question which seems to be open: Suppose that K is an infinite
(metrizable) compact space. Can Cw(K ) and Cp(K ) be homeomorphic? We show that the
answer is “no”, provided K is an infinite compact metrizableC-space. In particular our proof
works for any infinite compact metrizable finite-dimensional space K .
Keywords Function space · Pointwise convergence topology · Weak topology
Mathematics Subject Classification 46E10 · 54C35
For a compact space K we can consider three natural topologies on the set C(K ) of all con-
tinuous real-valued functions on K : the norm topology, the weak topology and the pointwise
topology. Let us denote C(K ) endowed with the latter two topologies by Cw(K ) and Cp(K )
respectively. Suppose that K is an uncountable compact space. Clearly, the space C(K )
equipped with the norm topology is homeomorphic neither to Cw(K ) nor to Cp(K ): indeed,
both Cw(K ) and Cp(K ) are not metrizable whereas the norm defines a metric on C(K ). For
a similar reason, if K is an infinite countable compact metrizable space, then Cw(K ) is not
homeomorphic to Cp(K ). In that case Cp(K ) is metrizable and Cw(K ) is not. If we try to
compare topologically Cw(K ) and Cp(K ), for an uncountable compact space K , the answer
is not obvious at all. There is a vast literature studying the weak and the pointwise topology
in function spaces, but surprisingly it seems to be unknown whether these two topologies are
homeomorphic. More precisely, we can address the following question: Let K be an uncount-
able compact (metrizable) space. Is it true that Cw(K ) and Cp(K ) are homeomorphic? This
M. Krupski was partially supported by the Polish National Science Center research grant
UMO-2012/07/N/ST1/03525.
B Mikołaj Krupski
mkrupski@mimuw.edu.pl
1 Institute of Mathematics, University of Warsaw, Ul. Banacha 2,
02-097 Warszawa, Poland
558 M. Krupski
question seems to be open even for standard uncountable metrizable compacta such as the
Cantor space 2ω or the unit interval [0, 1].
It was proved in [1] (cf. [4]) that if K is a finite-dimensional compactmetrizable space then
Cp(K ) and Cp([0, 1]ω) are not homeomorphic. On the other hand the celebrated Miljutin’s
theorem [6] asserts that for any two uncountable compact metrizable spaces K and L the
spaces Cw(K ) and Cw(L) are linearly homeomorphic. The combination of these two results
implies immediately that either Cp(2ω) is not homeomorphic to Cw(2ω) or Cp([0, 1]ω)
is not homeomorphic to Cw([0, 1]ω). Similarly, either Cp([0, 1]) is not homeomorphic to
Cw([0, 1]) or Cp([0, 1]ω) is not homeomorphic to Cw([0, 1]ω), and so on. It is however
unclear how to determine precisely which pairs of spaces are indeed not homeomorphic.
In this note we show that Cw(K ) and Cp(K ) are not homeomorphic for any infinite
compact metrizable C-space K (see Definition 1 below), in particular, for any infinite finite-
dimensional compact metrizable space K .
Our approach is based on some ideas from [5] (cf. [4,7]); however, to deal with the weak
topology on C(K ) we consider measures on the compact space K rather than points of that
space, as it was done in [4,5,7]. One of the key ingredients of the proof is Lemma 1 below.
Let K be a compact space. As usual, we identify the set C(K )∗, of all continuous linear
functionals on C(K ), with M(K )—the set of all signed Radon measures on K of finite
variation. Using this identification we can equip M(K ) with the weak* topology. For y ∈ K
we denote by δy ∈ M(K ) the corresponding Dirac measure. It is well-known that K can be
identified as the subspace {δy : y ∈ K } ⊆ M(K ). If A ⊆ M(K ) then Lin(A) is the linear
space spanned by A, i.e. the minimal linear subspace of M(K ) containing A.
We denote by ω the set of all non-negative integers, and N = ω\{0}. For a natural number
k we denote by [K ]≤k ([K ]<ω) the hyperspace of all at most k-element subsets of K (all
finite subsets of K ) equipped with the Vietoris topology.
Recall that sets of the form
OK
(
F; 1m
) = { f ∈ Cp(K ) : ∀x ∈ F | f (x)| < 1m
}
,
where F ∈ [K ]<ω and m ∈ N, are basic open neighborhoods of the function equal to zero
on K in Cp(K ).
Similarly, for a compact space L , if F is a finite subset of M(L) and n ∈ N, then
WL
(
F; 1n
) = { f ∈ Cw(L) : ∀μ ∈ F |μ( f )| < 1n
}
is a basic open neighborhood of the function equal to zero on L in Cw(L). If F = {x} or
F = {μ} we will write OK (x; 1m ), WL (μ; 1m ) rather than OK ({x}; 1m ), WL({μ}; 1m ).
For μ ∈ M(L) and n ∈ N we put
WL
(
μ; 1n
) = { f ∈ Cw(L) : |μ( f )| ≤ 1n
}
.
The mappings
πL : [K ]≤k × L → L
πK : [K ]≤k × L → [K ]≤k
are projections on L and [K ]≤k , respectively.
Similarly as in [4], for a fixed homeomorphism  : Cp(K ) → Cw(L) taking the zero
function on K to the zero function on L and for k,m, n ∈ N, we define the following sets:
Zk,m,n =
{
(E, y) ∈ [K ]≤k × L :  (OK
(
E; 1m
)) ⊆ WL
(
δy; 1n
)}
,
C(k,m, n) = πL(Zk,m,n).
On the weak and pointwise topologies in function spaces 559
The following proposition is easy to verify.
Proposition 1 The set Zk,m,n is a closed in [K ]≤k × L, for any k,m, n ∈ N.
Proof If (E, y) ∈ ([K ]≤k × L)\ Zk,m,n , then there is f ∈ Cp(K ) such that { f (x) : x ∈
E} ⊆ (− 1m , 1m ) and |δy(( f ))| = |( f )(y)| > 1n . Obviously, the set
{
F ∈ [K ]≤k : F ⊆ f −1(− 1m , 1m )
}
× {z ∈ L : |( f )(z)| > 1n
}
is an open neighborhood of (E, y) in [K ]≤k × L , disjoint from Zk,m,n . unionsq
It follows thatC(k,m, n) is closed in L (being a continuous image of a compact set). Note
that by the continuity of  we have L = ⋃k,m C(k,m, n). Now, for m, n, k ∈ N, we put
E(1,m, n) = C(1,m, n) and E(k,m, n) = C(k,m, n)\C(k − 1,m, n), for k > 1.
Clearly,
L =
⋃
k,m
E(k,m, n). (1)
For y ∈ E(k,m, n), let us put
E(y,m, n) = πK (π−1L (y) ∩ Zk,m,n),
i.e. E(y,m, n) is the family of all exactly k-element subsets E ⊆ K satisfying
(OK (E; 1m )) ⊆ WL(δy; 1n ) (this follows from the assumption y ∈ E(k,m, n)).
It is known that for any y ∈ E(k,m, n) the family E(y,m, n) is finite, cf. [9, Lemma
6.11.9]. Finally, let αm,n(y) = ⋃ E(y,m, n), for y ∈ E(k,m, n).
The following theorem is an immediate consequence of results proved in [5] (cf. [9,
Lemmas 6.11.10, 6.11.1])
Theorem 1 Suppose that K and L aremetrizable spaces. For any k,m ∈ N the set E(k,m, 1)
can be covered by countably many Gδ (in L) sets Gr such that for each r ∈ N, there are con-
tinuous mappings f ri : Gr → K, i = 1, . . . , pr , such that αm,1(y) = { f r1 (y), . . . , f rpr (y)}
for y ∈ Gr .
Proof In [5] an analogous result was proved under the assumption that  is a continuous
surjection of Cp(K ) onto Cp(L). Since the pointwise topology is weaker than the weak
topology and  : Cp(K ) → Cw(L) is a homeomorphism, it is also a continuous surjection
of Cp(K ) onto Cp(L). unionsq
By (1) sets of the formGr (considered for all k,m ∈ N) cover the whole space L . To avoid
multiple indexing, we abuse a little bit the notation here: To be more precise, we should write
Gk,m,1,r rather than Gr and consider a bijection of N3 onto N to reduce a number of indexes.
Since −1 : Cw(L) → Cp(K ) is continuous, for each x ∈ K and m ∈ N, there is
Fmx ∈ [M(L)]<ω and n ∈ N such that
−1
(
WL
(
Fmx ; 1n
)) ⊆ OK
(
x; 1m
)
. (2)
We will need the following lemma.
Lemma 1 If y ∈ E(k,m, 1) for some k,m ∈ N then δy ∈ Lin(⋃{Fmx : x ∈ αm,1(y)}).
560 M. Krupski
Proof Suppose that δy /∈ N = Lin(⋃{Fmx : x ∈ αm,1(y)}). By the definition of αm,1(y)
there is A ⊆ αm,1 such that

(
OK
(
A; 1m
)) ⊆ WL(δy; 1). (3)
Since δy /∈ N , we have
⋂
{Ker(μ) : μ ∈
⋃
{Fmx : x ∈ αm,1(y)}}  Ker(δy),
where Ker(ν) denotes the kernel of a functional ν (cf. [2, Lemma 3.9]). This means that
there is a continuous function g : L → R such that μ(δy) = g(y) = 0 and μ(g) = 0, for
any μ ∈ ⋃{Fmx : x ∈ αm,1(y)}. Scaling g if necessary, we have δy(g) = g(y) = 2 and
μ(g) = 0, for anyμ ∈ ⋃{Fmx : x ∈ A}. By (2), for every x ∈ A, we have |−1(g)(x)| < 1m ,
so −1(g) ∈ OK (A; 1m ). Therefore, by (3)
g = (−1(g)) ∈ WL(δy; 1).
This means that |g(y)| = |δy(g)| ≤ 1, a contradiction. unionsq
Definition 1 A normal space K is called a C-space if for any sequence of its open covers
(Ui )i∈ω, there exists a sequence (Vi )i∈ω of families of pairwise disjoint open sets such that
Vi is a refinement of Ui and
⋃
i∈ω Vi is a cover of K .
Definition 2 A family {(Ai , Bi ) : i ∈ ω} of pairs of disjoint closed subsets of a topological
space X is called essential if for every family {Li : i ∈ ω}, where Li is an arbitrary partition
between Ai and Bi for every i , we have
⋂
i∈ω Li = ∅. A normal space X is strongly infinite-
dimensional if it has an infinite essential family of pairs of disjoint closed sets.
It is well known that any finite-dimensional space, and more generally, any countable-
dimensional space (i.e. a space which is a countable union of finite-dimensional subspaces)
is a C-space. On the other hand, a strongly infinite-dimensional space is not a C-space. One
of the most natural examples of a strongly infinite-dimensional space is the Hilbert cube
[0, 1]ω.
Before we will proceed to the main result of this note, we need to make some preparatory
work concerning strongly infinite-dimensional spaces. Proposition 2 and Lemma 2 given
below are perhaps a part of folklore in the theory of infinite-dimension. Since we could
not find a proper reference in the literature, we shall enclose a proof here. The reasoning
presented below was communicated to the author by Roman Pol.
Lemma 2 Suppose that Z is a strongly infinite-dimensional compact metrizable space and
Y ⊆ Z is Gδ in Z. Then at least one of the following assertions holds true:
(a) Y contains a strongly infinite-dimensional compactum or
(b) Z\Y contains a strongly infinite-dimensional compactum.
Proof Since Y ⊆ Z is Gδ , we have Z\Y = ⋃∞k=1 Fk and each Fk is closed in Z (hence
compact). Fix an infinite essential family {(Ai , Bi ) : i ∈ ω} of pairs of disjoint closed subsets
of Z (witnessing the fact that Z is strongly infinite-dimensional). Let ω = ⋃∞k=0 Nk be a
partition of ω into infinite, pairwise disjoint sets.
Assume that (b) does not hold true. In particular, for each k ≥ 1 the set Fk is not strongly
infinite-dimensional and hence, by [9, Corollary 3.1.5] there is a sequence (Li )i∈Nk of par-
titions in Z between (Ai , Bi )i∈Nk with (
⋂
i∈Nk Li ) ∩ Fk = ∅.
On the weak and pointwise topologies in function spaces 561
We claim that
⋂∞
k=1
⋂
i∈Nk Li ⊆ Y is strongly infinite-dimensional [and hence (a) holds].
Indeed, otherwise there is a sequence (Li )i∈N0 of partitions in Z between (Ai , Bi )i∈N0
with
⎛
⎝
⋂
i∈N0
Li
⎞
⎠ ∩
∞⋂
k=1
⋂
i∈Nk
Li = ∅,
which is a contradiction with our assumption that the family {(Ai , Bi ) : i ∈ ω} is essential.
unionsq
Proposition 2 Suppose that X is a strongly infinite-dimensional compact metrizable space.
Let X = ⋃n∈ω Xn, where each Xn is a Gδ subset of X. Then, there is n ∈ ω such that Xn
contains a strongly infinite-dimensional compactum.
Proof Striving for a contradiction assume that none of Xn’s contains a strongly infinite-
dimensional compactum.By induction,we construct a decreasing sequence F0 ⊇ F1 ⊇ · · · ⊇
Fn ⊇ . . . of strongly infinite-dimensional compacta such that, for each i ∈ ω, Fi ⊆ X\Xi .
For n = 0 we apply Lemma 2 with Z = X and Y = X0. By our assumption (a) does not
hold and hence there is a strongly infinite dimensional compactum F0 ⊆ X\X0.
Assume that, for n ∈ ω, we already constructed a sequence F0 ⊇ · · · ⊇ Fn of strongly
infinite-dimensional compacta such that Fi ⊆ X\Xi . We apply Lemma 2 with Z = Fn and
Y = Xn+1 ∩ Fn . Again, by our assumption (a) does not hold and consequently there exists a
strongly infinite dimensional compact set Fn+1 ⊆ Fn\(Xn+1 ∩ Fn). This ends the inductive
construction.
Since (Fn)n∈ω is a decreasing sequence of non-empty compact sets, it has a non-empty
intersection
⋂
n∈ω Fn . On the other hand
⋂
n∈ω Fn ⊆ X\
⋃
n∈ω Xn = ∅, a contradiction. unionsq
Finally, we can prove the following.
Theorem 2 If K is a compact metrizable C-space, then Cp(K ) and Cw([0, 1]ω) are not
homeomorphic.
Proof Otherwise, there is a homeomorphism  : Cp(K ) → Cw([0, 1]ω). Since function
spaces are homogeneous, we can without loss of generality assume that  takes the zero
function on K to the zero function on [0, 1]ω. By Theorem 1 we have [0, 1]ω = ⋃r∈N Gr ,
where each Gr is a Gδ subset of [0, 1]ω and for every r ∈ N, there are continuous mappings
f ri : Gr → K , i = 1, . . . , pr , such that αm,1(y) = { f r1 (y), . . . , f rpr (y)} for y ∈ Gr .
By Proposition 2 there is r ∈ N such that Gr contains a strongly infinite-dimensional
compactum Q ⊆ Gr .
Let f = i≤pr ( f ri  Q) : Q → K pr be the restriction to Q of the diagonal mapping, i.e.
f (y) = ( f r1 (y), . . . , f rpr (y)), for y ∈ Q ⊆ Gr .
Since K pr is a C-space (cf. [8]) and Q is not, not all fibers of f are zero-dimensional (in
fact not all of them are C-spaces), cf. [3, 5.4].
Hence, there is x = (x1, . . . , xpr ) ∈ K pr such that f −1(x) is uncountable. Note that for
any y ∈ f −1(x) we have αm,1(y) = {x1, . . . , xpr }.
Consider
Fx =
pr⋃
i=1
Fmxi .
562 M. Krupski
Obviously this set is finite. For μ ∈ M(L) let us put Aμ = {y ∈ L : μ({y}) = 0}. For
each μ ∈ M(L) the set Aμ is countable being the set of atoms of a measure. From Lemma 1
it follows that for each y ∈ f −1(x) there is μ ∈ Fx such that y ∈ Aμ.
This means that
f −1(x) ⊆
⋃
μ∈Fx
Aμ.
However, the latter set is countable and thus cannot cover the uncountable fiber f −1(x), a
contradiction unionsq
Combining the above theorem with the Miljutin’s theorem [6] we get the following.
Corollary 1 If K is an uncountable compact metrizable C-space then Cw(K ) and Cp(K )
are not homeomorphic.
In particular, the above corollary covers the important case of all uncountable finite-
dimensional compacta.
Open questions
ThoughCorollary 1 is quite general, our method does not work for all uncountablemetrizable
compacta. Thus we do not know the answer to the following basic question mentioned in the
Introduction.
Question 1 Suppose that K is an uncountable compact metrizable space (which is not a
C-space). Is it true that Cp(K ) and Cw(K ) are not homeomorphic?
It seems that the most interesting particular case of the above question is the following:
Question 2 Is it true that Cp([0, 1]ω) and Cw([0, 1]ω) are not homeomorphic?
Although we have the proof that, for example,Cw(2ω) andCp(2ω) are not homeomorphic
our method seems to be fairly complicated. Moreover it does not provide any topological
property distinguishing Cp(2ω) and Cw(2ω). Thus the following problem seems to be inter-
esting.
Problem 1 Find a topological property distinguishing Cp(2ω) and Cw(2ω). Find a topolog-
ical property distinguishing Cp([0, 1]) and Cw([0, 1]).
It is reasonable to ask also what happens outside the metrizable case:
Problem 2 Is it true that Cp(K ) and Cw(K ) are not homeomorphic for any infinite compact
space K ?
Acknowledgments The author is indebted to Witold Marciszewski for valuable comments and remarks.
This research was partially supported by the Polish National Science Center research grant UMO-
2012/07/N/ST1/03525.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and
reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons license, and indicate if changes were made.
On the weak and pointwise topologies in function spaces 563
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