Cp(X) IS NEVER WEAKLY PSEUDOCOMPACT

F. HERNANDEZ-HERNANDEZ, R. ROJAS-HERNANDEZ, AND A. TAMARIZ-MASCARUA

Abstract. A space Z is weakly pseudompact if Z is Gδ-dense in at least one of its compactifications.F.W. Eckertson asked in 1996 [6] if the spaces Rω1 , Nω1 or Sω1 are weakly pseudocompact where S isthe Sorgenfrey line. In [5] a more general question was posed: What conditions must spaces X and Gsatisfy so that the space Cp(X, G) be weakly pseudocompact?

In this paper we answer Eckertson’s question in the negative and we prove that Cp(X, G) is neverweakly pseudocompact if G is a metrizable separable locally compact non compact topological group,assuming that Cp(X, G) is dense in GX . In particular, Cp(X) is not weakly pseudocompact for anyTychonoff space X, and Cp(X, Z) is not weakly pseudocompact if X is zero-dimensional T2.

1. Notations, basic definitions and introduction

Throughout this article all topological spaces are considered Tychonoff and with more than one pointif the contrary is not specified. As usual, the real line with the Euclidean topology is denoted by R, andits subspace of integers by Z. The symbol ω denotes the first infinite cardinal number and N the naturalnumbers; that is, N = ω \ {0}. The first uncountable cardinal number is ω1. For a space X and a subsetH of X, clX H and intXH will mean the closure and the interior of H in the space X. If there is nodoubt as to what space X we are considering, we will simply write clH and intH instead of clX H andintXH. For a set Y , [Y ]<ω, [Y ]≤ω and [Y ]ω signify the collection of finite subsets, countable subsets andinfinite countable subsets of Y , respectively.

We are going to denote by Cp(X,Y ) the space of continuous functions from X to Y with the topologyof the pointwise convergence, which is that inherited by the product topology in Y X ; that is, the topologyin Cp(X,Y ) is the topology generated by all the sets of the form:

[x1, . . . , xn;B1, . . . , Bn] = {f ∈ Cp(X,Y ) : f(xi) ∈ Bi, i = 1, . . . , n}

where n ∈ ω, {x1, . . . , xn} ⊆ X, and B1, . . . , Bn are open subsets of Y. When Y is the real line with itsusual topology, we write Cp(X) instead of Cp(X,R). For every space of the form Cp(X,Y ) considered inthis article, the spaces X and Y are such that Cp(X,Y ) is dense in Y X .

A compactification of a spaceX is a compact spaceK containingX as a dense subspace. The statement“X ⊆ Y is Gδ-dense in Y ” means that each nonempty Gδ-set in Y contains at least one point in X.

A space X is weakly pseudocompact if it is Gδ-dense in some of its compactifications. Every pseudo-compact space is weakly pseudocompact; weak pseudocompactness is a productive property; and everyweakly pseudocompact space is a Baire space (see [8]). F.W. Eckertson posed back in 1996 in [6] thefollowing problems:

Problem 1.1. (1) Is Rκ, Zκ or Sκ weakly pseudocompact when S is the Sorgenfrey line and κ > ω?(2) Is a Σ-product of Rκ, Zκ or Sκ weakly pseudocompact?

A more general problem is:

Problem 1.2. Characterize spaces X and Y for which the space of continuous functions from X to Ywith the pointwise convergence topology, Cp(X,Y ), is weakly pseudocompact.

Some contributions in the direction of solving this problem are made in [2], [3], [4], [5] and [9]. In thisarticle, we continue these efforts and prove the following:

2000 Mathematics Subject Classification. Primary: 54C35, 54D35, 54C45. Secondary: 54B10, 54E52.Key words and phrases. Space of continuous functions; weakly pseudocompact spaces; Σ-product; limits of inverse systems;nowhere dense subsets of compact spaces.Research of the first-named author was supported by CONACYT Grant 169078-F.The second-named author was supported for a postdoctoral position by DGAPA-UNAM.Research of the third-named author was supported by CONACYT Grant 220426 and by PASPA-DGAPA-UNAM.

1

2 F. HERNANDEZ-HERNANDEZ, R. ROJAS-HERNANDEZ, AND A. TAMARIZ-MASCARUA

Theorem 1.3. Let G be a metrizable separable locally compact and non compact topological group. Then,the space Cp(X,G) is not weakly pseudocompact.

Numerous outstanding consequences arise from this result. In particular we obtain a negative answerto items (1) and (2) in Problem 1.1, and determine a wide range class of topological spaces X and G forwhich Problem 1.2 has a negative answer. Furthermore, we show that for every cardinal number κ, thereis an R-factorizable topological group with weight κ · ω which is Sanchez-Okunev countably compact,Oxtoby countably compact and non weakly pseudocompact, answering a part of a question posed in [4].

Section 2 below is devoted to presenting the concepts and first lemmas that we are going to use inorder to prove Theorem 1.3. In Section 3, we give some properties about nowhere dense subsets in thelimit of an inverse system of compact spaces. In Section 4 we give some base assignments for the spacesintroduced in Section 2. In Section 5 we discuss the relations given by a condensation c : S → X betweenthe inverse systems defined by S and X as constructed in Section 2. Section 6 is devoted to proving ourmain result, and some of its consequences are listed in Section 7. Finally, in Section 8, we present someopen problems.

2. First notations, definitions and lemmas

Let X be a Tychonoff space and let Y be an infinite set. Assume that D ⊆ A ⊆ Y . We will denoteby pD,A the projection from XA onto XD. Let πD,A : β(XA) → β(XD) be the continuous extensionof pD,A. We set P =

∏G∈[Y ]≤ω β(XG). For each A ∈ [Y ]≤ω, pA is the projection from P onto β(XA).

Consider the diagonal map d = ∆G∈[Y ]≤ωpG,Y : XY → P and let W = d(XY ). Moreover, set

K := {f ∈ P : πD,A(pA(f)) = pD(f) whenever D ⊆ A ∈ [Y ]≤ω}.Finally put qA = pA �K for all A ∈ [Y ]≤ω.

Remark 2.1. Assume that C ⊆ D ⊆ A ∈ [Y ]≤ω. Then we have the following equalities:(a) πD,A ◦ qA = qD;(b) πC,A = πC,D ◦ πD,A; and(c) pA,Y = qA ◦ d.

Proof. (a) If f ∈ K, then πD,A(qA(f)) = πD,A(pA(f)) = pD(f) = qD(f). Therefore πD,A ◦ qA = qD.(b) Since the functions pC,A, pC,D and pD,A are projections, we have that pC,A = pC,D◦pD,A. Moreover,

πC,A, and πC,D ◦πD,A are extensions of pC,A and pC,D ◦pD,A from XA on β(XA), respectively. Therefore,πC,A = πC,D ◦ πD,A.

(c) For each f ∈ XY observe that qA ◦ d(f) = qA(d(f)) = pA(d(f)) = pA,Y (f). So, pA,Y = qA ◦ d. �

XY

pA,Y

��

d // W� � //

qA �Wzzuuuuuuuuuuuu K� _

��

qA

{{vvvvvvvvvvv

XA � � //

pD,A

��pC,A

��

β(XA)

πD,A

��πC,A

��

PpAoo

XD � � //

pC,D

��

β(XD)

πC,D

��XC � � // β(XC)

Diagram 1. All the circles are commutative

Remark 2.2. Because of the above mentioned definitions and Remark 2.1.(b), the triplet

M = {β(XA), πD,A, [Y ]≤ω}is an inverse system, where [Y ]≤ω is directed by the relation ⊆. Besides, K is the limit of the inversesystem M. (See [7], page 98.)

Cp(X) IS NEVER WEAKLY PSEUDOCOMPACT 3

Remark 2.3. The space K is compact.

Proof. Since K is the limit of the inverse system M and β(XA) is Hausdorff for each A ∈ [Y ]≤ω, thespace K is closed in P (see [7, Proposition 2.5.1]). Thus, K is compact. �

Lemma 2.4. The family of all sets of the form q−1A (U) where A ∈ [Y ]≤ω and U is open in β(XA) is a

base for the topology on K.

Proof. Let f ∈ K, and let O be a canonical open set in P containing f . Then, there are n < ω,G0, . . . , Gn ∈ [Y ]≤ω and open subsets B0, . . . , Bn of β(XG0), . . . , β(XGn), respectively, such that O =⋂

i≤n p−1Gi

(Bi). Take A =⋃

i≤nGi and consider the open subset U :=⋂

i≤n π−1Gi,A

(Bi) of β(XA).We claim that f ∈ q−1

A (U) ⊆ O ∩ K. Indeed, πGi,A(qA(f)) = qGi(f) = pGi(f) ∈ Bi for each i ≤ n

and then f ∈ q−1A (U). Now, by using Remark 2.1 we obtain that q−1

A (U) = q−1A (

⋂i≤n π

−1Gi,A

(Bi)) =⋂i≤n q

−1A (π−1

Gi,A(Bi)) =

⋂i≤n q

−1Gi

(Bi) =⋂

i≤n p−1Gi

(Bi) ∩K = O ∩K. �

Lemma 2.5. The family {pG,Y : G ∈ [Y ]≤ω} separates points as well as points from closed sets in XY .

Proof. Since XY is T0, it is enough to prove that {pG,Y : G ∈ [Y ]≤ω} separates points from closed sets.Pick a point f and a closed set F such that f 6∈ F . Choose a canonical open set O :=

⋂i≤n p

−1yi,Y

(Bi)in XY such that f ∈ O and O ∩ F = ∅. Take G = {y0, . . . , yn}. Note that pG,Y (f) ∈ pG,Y (O). Thefact that p−1

G,Y (pG,Y (O)) = O implies that p−1G,Y (pG,Y (O)) ∩ F = ∅ and hence pG,Y (O) ∩ pG,Y (F ) = ∅.

Therefore pG,Y (f) 6∈ cl(pG,Y (F )). �

Because of Theorem 2.3.20 in [7] and Lemma 2.5, we obtain:

Corollary 2.6. The map d : XY →W is an embedding.

Remark 2.7. The space K contains W as a dense subspace.

Proof. First of all we have to prove that W ⊆ K. Pick g ∈ W . Choose f ∈ XY such that g = d(f).Since πD,A(pA(g)) = πD,A(pA(d(f))) = πD,A(pA,Y (f)) = pD,A(pA,Y (f)) = pD,Y (f) = pD(d(f)) = pD(g)whenever D ⊆ A ∈ [Y ]≤ω, we obtain that g ∈ K.

Because of Lemma 2.4, to show that W is dense in K it is enough to verify that W ∩ q−1A (U) 6= ∅

for each A ∈ [Y ]≤ω and every open subset U of β(XA). Pick A ∈ [Y ]≤ω and an open subset U ofβ(XA). Select f ∈ p−1

A,Y (U ∩ XA) ⊆ XY . Then qA(d(f)) = pA,Y (f) ∈ U ∩ XA ⊆ U and therefored(f) ∈W ∩ q−1

A (U). �

Remark 2.8. For each A ∈ [Y ]≤ω the function qA : K → β(XA) is onto.

Proof. Since qA(W ) = qA(d(XY )) = pA,Y (XY ) = XA, the continuity of qA and the compactness of Kimply that the function qA maps K onto β(XA). �

Lemma 2.9. If XY has countable cellularity and C ⊆ XY is Gδ-dense, then K is the Stone-Cechcompactification of d(C).

Proof. We will prove that K satisfies the extension property which characterizes β(d(C)). Let φ : d(C) →I be a continuous map, where I is the unit interval. By the Tkachenko Factorization Theorem [1, Problem1.7.B], there exists A ∈ [Y ]≤ω and a map φA : pA,Y (C) → I such that φ ◦ d �C= φA ◦ pA,Y �C . LetψA : β(pA,Y (C)) → I be the continuous extension of φA. By hypothesis C is Gδ-dense in XY , soβ(pA,Y (C)) = β(XA) = qA(K). Thus the composition ψA ◦ qA : K → I makes sense. Let us verify thatthe map ψA ◦ qA is the desired continuous extension of φ. Indeed, given f ∈ d(C) select g ∈ C such thatf = d(g) and observe that

ψA ◦ qA(f) = ψA ◦ qA ◦ d(g) = ψA ◦ pA,Y (g) = φA ◦ pA,Y (g) = φ ◦ d(g) = φ(f).

Therefore, K is the Stone-Cech compactification of d(C). �

3. Nowhere dense subsets of K

Denote by N the family of all nowhere dense subsets of the inverse limit K. We are going to obtainsome lemmas which will play a crucial role when we prove the main theorem of this article (Theorem 6.1,below). These lemmas have to do with some relations concerning to the family N .

Remark 3.1. Let φ : K → L be a continuous map and let Z be a dense subspace of K such that φ �Z

is an embedding. Then φ−1(φ(N )) ⊆ N .

4 F. HERNANDEZ-HERNANDEZ, R. ROJAS-HERNANDEZ, AND A. TAMARIZ-MASCARUA

Proof. Fix N ∈ N . We can suppose that N is closed in K. Let U be the interior of the closed setφ−1(φ(N)) in K. Assume that z ∈ Z ∩ U . Since φ(z) ∈ φ(N), there is x ∈ N such that φ(z) = φ(x),and hence z = x ∈ N (see p. 168 in [7, Lemma 3.5.6]). As z was arbitrary, it must happen thatU ⊆ cl(Z ∩ U) ⊆ N . It follows that U = ∅. Therefore φ−1(φ(N)) ∈ N . �

Lemma 3.2. Suppose that φ : K → L is a quotient map and Z is a dense subspace of K such that φ �Z

is an embedding. If U is open in K, then there exists N ∈ N such that U ∪N is closed and φ-saturatedin K.

Proof. Note that since φ �Z is an embedding, we must have φ−1(φ(x)) = {x} for each x ∈ Z.Since φ(U ∩ Z) is open in φ(Z), we can find an open set O in L such that O ∩ φ(Z) = φ(U ∩ Z).

Consider the φ-saturated and open set V = φ−1(O). Note that V ∩ Z = U ∩ Z. Indeed, the relationU ∩ Z ⊆ V ∩ Z is obvious. Now if f ∈ V ∩ Z then φ(f) ∈ O ∩ φ(Z) = φ(U ∩ Z). This means that thereis g ∈ U ∩ Z such that φ(g) = φ(f). The fact that g ∈ Z implies that g = f ; so, f ∈ U ∩ Z. Now, notethat the equality V ∩ Z = U ∩ Z implies that cl(U) = cl(V ) = cl(U ∩ V ).

We claim that the set N = φ−1(φ(cl(U) \ (U ∩ V ))) is as we need. On the one hand, note thatcl(U) \ (U ∩ V ) = cl(U ∩ V ) \ (U ∩ V ) ∈ N . It follows from Remark 3.1 that N ∈ N . On the other hand,note that

U ∪N ⊆ φ−1(φ(cl(U))) = φ−1(φ(cl(V ))) = φ−1(φ(V ∪ (cl(V ) \ V )) = φ−1(φ(V )) ∪ φ−1(φ(cl(V ) \ V )) =V ∪ φ−1(φ(cl(U) \ V )) ⊆ V ∪N ⊆ cl(U) ∪N ⊆ U ∪N.

Therefore U ∪N = φ−1(φ(cl(U))) is closed and φ-saturated in K. �

Lemma 3.3. Assume that for each A ∈ [Y ]≤ω there exist NA ∈ [N ]≤ω and BA ∈ [τ(β(XA))]≤ω satisfyingthe following conditions:(a) NA =

⋃F∈[A]<ω NF ;

(b) BA =⋃

F∈[A]<ω π−1A,F (BF ) and BA is a π-base for β(XA); and

(c) BK =⋃

F∈[Y ]<ω q−1F (BF ) is a π-base for K.

Then there exists A ∈ [Y ]ω such that qA(N) is nowhere dense in β(XA) for each N ∈ NA.

Proof. For each B ∈ BK fix FB ∈ [Y ]<ω such that B ∈ q−1FB

(BFB). Let M be a countable elementary

submodel of H(θ) for some large enough cardinal θ which contains everything relevant for this proof, inparticular the assignments from the hypotheses. Consider the set A =

⋃(M∩ [Y ]≤ω) ∈ [Y ]ω. Note that

q−1A (BA) =

⋃F∈[A]<ω q

−1A (π−1

A,F (BF )) =⋃

F∈[A]<ω q−1F (BF ) = BK ∩M. Choose N ∈ NA. The clause (a)

implies that N ∈ NF for some F ∈ [A]<ω. It follows that N ∈ N ∩M. By using the conditions (b) and(c) we obtain that

∀U ∈ BK∃V ∈ BK(V ⊆ U ∧ V ∩N = ∅) ⇔ ∀U ∈ (BK ∩M)∃V ∈ (BK ∩M)(V ⊆ U ∧ V ∩N = ∅)⇔ ∀U ∈ q−1

A (BA)∃V ∈ q−1A (BA)(V ⊆ U ∧ V ∩N = ∅)

⇔ ∀U ∈ BA∃V ∈ BA(V ⊆ U ∧ V ∩ qA(N) = ∅).From these equivalences we conclude that qA(N) is nowhere dense in β(XA). �

Remark 3.4. Let e : S → X be a continuous onto map and let B be a family of open subsets of X suchthat e−1(B) is a π-base for S. If N is nowhere dense in S, then e(N) is nowhere dense in X.

Proof. Let N be a nowhere dense set in S. Choose an arbitrary nonempty open subset U of X. By ourhypothesis, the open set e−1(U) contains a member e−1(B) of e−1(B) such that e−1(B) ∩N = ∅. Sincethe map e is onto, we must have that B ⊆ U . Besides, it is clear that B ∩ e(N) = ∅. In this way, e(N)is nowhere dense in X. �

4. The spaces W and K when X is locally compact and non compact

In this section we determine external bases for W in K and XA in βXA, obtained from a given baseof X, for the case that X is locally compact and non compact. First we recall some basic facts.

Remark 4.1. Let D be a dense subspace of X and let B be a collection of open subsets of X. If B ∩Dis a base (resp., a π-base) for D, then B contains a local base for each point of D in X (resp., is a π-basefor X).

Proof. Select d ∈ D. Let U be an open set in X containing d. Choose an open set O in X such thatd ∈ O ⊆ clX O ⊆ U . It happens that there is an element B in B such that d ∈ B ∩ D ⊆ O. Then,d ∈ B ⊆ clX B = clX(B ∩D) ⊆ clX O ⊆ U . In the same way we can prove the case of π-bases. �

Cp(X) IS NEVER WEAKLY PSEUDOCOMPACT 5

Remark 4.2. Let c : S → X be a condensation and let e : βS → βX be the continuous extension of c.If B is a family of open subsets of βX and c−1(B ∩X) is a π-base for S, then e−1(B) is a π-base for βS.

Proof. Given an open set B in βX note that e−1(B)∩S = c−1(B)∩S = c−1(B∩X). Hence e−1(B)∩S =c−1(B ∩X). It follows that e−1(B)∩S = c−1(B ∩X) is a π-base for S. Therefore, we can apply Remark4.1 to see that e−1(B) is a π-base for βS. �

Assume that the space X is locally compact and non compact. Fix a base B for the space X constitutedby open subsets with compact closure. For each F ∈ [Y ]<ω and s ∈ BF we set s =

⋂y∈F p

−1y,F (s(y)),

where py,F : XF → X is the y-projection. Given F ∈ [Y ]<ω and s ∈ BF note that since the closure ofeach element of B is compact, the set s has a compact closure in XF and is open in β(XF ); and as aconsequence π−1

F,A(s) and q−1F (s) are open in β(XA) and K, respectively, whenever F ⊆ A ⊆ Y . Consider

the next families of open sets:

BK = {q−1F (s) : F ∈ [Y ]<ω and s ∈ BF }

andBA = {π−1

F,A(s) : F ∈ [A]<ω and s ∈ BF },for each A ∈ [Y ]≤ω.

Remark 4.3. Suppose that A ∈ [Y ]≤ω. Then we have the following:(a) If F ∈ [A]<ω and s ∈ BF , then q−1

F (s) = q−1A (π−1

F,A(s)).(b) BA =

⋃F∈[A]<ω π

−1A,F (BF ).

(c) BK =⋃

F∈[Y ]<ω q−1F (BF ).

Lemma 4.4. (a) The family BA contains a local base at each point of XA in β(XA), for all A ∈ [Y ]≤ω.(b) The collection BK contains a local base for every point of W in K.

Proof. (a) Observe that the family {p−1F,A(s) : F ∈ [A]<ω and s ∈ BF } is a base of XA. Since π−1

F,A(s) ∩XA = p−1

F,A(s) for each F ∈ [A]<ω and s ∈ BF , from Remark 4.1 we conclude that BA contains a localbase at each point of XA in β(XA).

(b) Since the family {p−1F,Y (s) : F ∈ [Y ]<ω and s ∈ BF } is a base for XY and d is an embedding, the

collection {d(p−1F,Y (s)) : F ∈ [Y ]<ω and s ∈ BF } is a base for d(XY ) = W . By using Remark 2.1.(c), we

can see that q−1F (s) ∩W = (qF �W )−1(s) = d(p−1

F,Y (s)) for each F ∈ [Y ]<ω and s ∈ BF . Then we canapply Remark 4.1 to see that the collection BK contains a local base for every point of W in K. �

5. Inverse systems and condensations

Let c : S → X be a condensation, that is a continuous and bijective map, from a space S onto thelocally compact non compact space X. We will suppose that the base B of X, constituted by open subsetswith compact closure, satisfies that c−1(B) is a π-base for the topology of S.

In this section we will use the condensation c to relate the topological elements defined in Section 2for the space S with those defined for the space X. Each of these elements will be escorted by an S as asubscript or superscript when we move in the S-world. In particular,

MS = {β(SA), πSD,A, [Y ]≤ω}

is the inverse system associated to the space SY and KS is the limit of the inverse system MS . We willhighlight some natural relations determined by c between the two topological structures defined by theinverse systems MS and M.

For each A ⊆ Y , note that the product map cA := cA : SA → XA is also a condensation. LeteA : β(SA) → β(XA) be the continuous extension of cA for every A ⊆ Y . We will denote by EY theproduct function

∏G∈[Y ]≤ω eG : PS → P . Moreover, we set ξY = EY � KS and ζY = EY � WS .

Remark 5.1. (a) The family {(pSF,A)−1(c−1

F (s)) : F ∈ [A]<ω and s ∈ BF } is a π-base for SA.(b) The collection {dS((pS

F,Y )−1(c−1F (s))) : F ∈ [A]<ω and s ∈ BF } is a π-base for WS .

Lemma 5.2. Assume that F ⊆ A ⊆ Y , then we have the following equalities:(a) cF ◦ pS

F,A = pF,A ◦ cA;(b) eF ◦ πS

F,A = πF,A ◦ eA;(c) ζY ◦ dS = d ◦ cY ; and(d) eA ◦ qS

A = qA ◦ ξY whenever A ∈ [Y ]≤ω.

6 F. HERNANDEZ-HERNANDEZ, R. ROJAS-HERNANDEZ, AND A. TAMARIZ-MASCARUA

Proof. (a) For each f ∈ SA and y ∈ F we have that cF ◦ pSA,F (f)(y) = (cF (f �F ))(y) = c(f �F (y)) =

c(f(y)) = (cA(f))(y) = (cA(f) �F )(y) = pA,F ◦ cA(f)(y). Hence cF ◦ pSA,F = pA,F ◦ cA.

(b) Note that eF ◦ πSF,A �SA= cF ◦ pS

F,A = pF,A ◦ cA = πF,A ◦ eA �SA . Since SA is dense in β(SA), wemust have that eF ◦ πS

F,A = πF,A ◦ eA.

(c) Let us observe that ζY ◦ dS = EY ◦ dS =∏

G∈[Y ]≤ω cG ◦ ∆G∈[Y ]≤ωpSG,Y = ∆G∈[Y ]≤ωcG ◦ pS

G,Y =∆G∈[Y ]≤ωpG,Y ◦ cY = d ◦ cY .

(d) Given dS(f) ∈WS , it happens that eA ◦qSA(dS(f)) = eA ◦pS

A,Y (f) = cA ◦pSA,Y (f) = pA,Y ◦ cY (f) =

qA ◦ d ◦ cY (f) = qA ◦ ζY ◦ dS(f) = qA ◦ ζY (dS(f)) = qA ◦ ξY (dS(f)). Since WS is dense in KS , we musthave that eA ◦ qS

A = qA ◦ ξY . �

SY

dS

��

pSA,Y

��

cY

((XY

d��

pA,Y

��

WS� � //

qSA �W

��

ζY

((KS� _

��qSA

��

ξY // K_�

��qA

��

W?_oo

qA �W

��

PSEY //

pSA

��

P

pA

��SA � � //

pSF,A

��

β(SA)eA //

πSF,A

��

β(XA)

πF,A

��

XA? _oo

pF,A

��SF � � //

cF

66β(SF )eF // β(XF ) XF? _oo

Diagram 2. All the circles are commutative

From Lemma 5.2.(c) we deduce that ζY = d ◦ cY ◦ d−1S : WS → W is a condensation. Since KS is

compact, WS is dense in KS , W is dense in K, and ξY � WS = ζY , we must have that ξY maps KS

continuously onto K. We consider the families BSK = ξ−1

Y (BK) and BSA = e−1

A (BA), for each A ∈ [Y ]≤ω.

Lemma 5.3. (a) BSA is a π-base for β(SA) and BS

A =⋃

F∈[A]<ω (πSF,A)−1(BS

F ) for every A ∈ [Y ]≤ω.(b) BS

K is a π-base for KS and BSK =

⋃F∈[Y ]<ω (qS

F )−1(BSF ).

Proof. (a) Note that for each B = π−1F,A(s) ∈ BA we have the equality c−1

A (B ∩ XA) = c−1A (π−1

F,A(s) ∩XA) = c−1

A (p−1F,A(s)) = (pS

F,A)−1(c−1F (s)). Then the family c−1

A (BA ∩ XA) = {(pSF,A)−1(c−1

F (s)) : F ∈[A]<ω and s ∈ BF } is a π-base for SA. By applying Remark 4.2 we conclude that BS

A is a π-base forβ(SA). To verify the second part of our statement note that BS

A = e−1A (BA) = e−1

A (⋃

F∈[A]<ω π−1F,A(BF )) =⋃

F∈[A]<ω e−1A (π−1

F,A(BF )) =⋃

F∈[A]<ω (πSF,A)−1(e−1

F (BF )) =⋃

F∈[A]<ω (πSF,A)−1(BS

F ).

(b) Observe that for every B = q−1F (s) ∈ BK we have the equality ζ−1

Y (B ∩ W ) = ζ−1Y (q−1

F (s) ∩W ) = ζ−1

Y (d(p−1F (s))) = dS(c−1

Y (p−1F (s))) = dS((pS

F,Y )−1(c−1F (s))). Then the family ζ−1

Y (BK ∩ W ) ={dS((pS

F,Y )−1(c−1F (s))) : F ∈ [A]<ω and s ∈ BF } is a π-base for WS . By applying Remark 4.2 we

conclude that BSK is a π-base for KS . To verify the second part of this statement note that BS

K =ξ−1Y (BK) = ξ−1

Y (⋃

F∈[Y ]<ω q−1F (BF )) =

⋃F∈[Y ]<ω ξ

−1Y (q−1

F (BF )) =⋃

F∈[Y ]<ω (qSF )−1(e−1

F (BF )). �

6. Weak pseudocompactness and Gδ-dense subspaces of XY

In this section, we prove the main result of our article:

Cp(X) IS NEVER WEAKLY PSEUDOCOMPACT 7

Theorem 6.1. Let C be a Gδ-dense subspace of SY and let c : S → X be a condensation. Assume thatX is not compact and that it has a countable base B such that: cl(B) is compact for all B ∈ B, andc−1(B) is a π-base for S. Then the space C is not weakly pseudocompact.

Proof. Since weak pseudocompactness is productive and the product of Gδ-dense subspaces is a Gδ-densesubspace of the product, we may assume that |Y | ≥ ω.

We will use the constructions from sections 1, 2 and 3. Let Z = dS(C). By Remark 2.6 it is enoughto show that Z is not weakly pseudocompact. Let L be an arbitrary compactification of Z. By Remark2.9, there exists a continuous map φ : KS → L such that φ �Z is an embedding. We must show that φ(Z)is not Gδ-dense in L.

The map φ being closed is a quotient map. According to Lemma 3.2 for each open subset U of Kwe can fix a nowhere dense subset NU of KS such that U ∪ NU is closed and φ-saturated in KS . Foreach A ∈ [Y ]≤ω let N S

A = {NU : U ∈ (qSA)−1(BS

A)}. Note that N SA =

⋃F∈[A]≤ω N S

F for all A ∈ [Y ]≤ω.Because of Lemmas 3.3 and 5.3 we can find an A ∈ [Y ]ω such that qS

A(N SA) is a family of nowhere dense

subsets of β(SA). By applying Remark 3.4 and Lemma 5.3 we deduce that eA(qSA(N S

A)) is a family ofnowhere dense subsets of β(XA). Since eA(qS

A(N SA)) is countable and β(XA) has the Baire property, the

set NA =⋃eA(qS

A(N SA)) is nowhere dense in β(XA).

Let DA = β(XA) \XA. Note that the space XA is nowhere locally compact because A is infinite andX is not compact. As a consequence, the set DA is dense in β(XA). It follows that DA \NA 6= ∅. Fix amap fA ∈ DA \NA. By Lemma 4.4, for each g ∈ XA we can find Bg ∈ BA such that g ∈ Bg and fA 6∈ Bg.The space XA being second countable has the Lindelof property. Since the family {Bg : g ∈ XA} coversXA, it contains a countable subfamily CA which covers XA. Note that CA ⊆ BA and fA 6∈

⋃CA.

We are ready to show that φ(Z) is not Gδ-dense in L. Observe that G = {β(SY ) \ (U ∪ NU ) : U ∈(qS

A)−1(e−1A (CA))} is a countable family of open φ-saturated subsets of KS and, as a consequence, φ(G)

is a countable family of open subsets of L.

Claim. The Gδ-set⋂φ(G) is nonempty and misses φ(Z).

Proof of the claim. For the proof of the first part of the Claim, fix f ∈ (qSA)−1(e−1

A (fA)). Pick U ∈(qS

A)−1(e−1A (CA)) arbitrarily and choose B ∈ CA such that U = (qS

A)−1(e−1A (B)). From B ∈ CA ⊆ BA

we deduce that fA 6∈ B = eA(qSA(U)). Besides U ∈ (qS

A)−1(e−1A (BA)) = (qS

A)−1(BSA) and in this way

NU ∈ N SA . We know that fA 6∈ NA and in particular fA 6∈ eA(qS

A(NU )). From these facts we deduce thatfA 6∈ eA(qS

A(U ∪ NU )). Hence f ∈ β(SY ) \ (U ∪ NU ). Since U is arbitrary, we conclude that f ∈⋂G.

Therefore φ(f) ∈⋂φ(G).

For the proof of the second part of the Claim, note that (⋂G) ∩

⋃(qS

A)−1(e−1A (CA)) = ∅. Then

qSA(

⋂G) ∩ e−1

A (⋃CA) = ∅. The fact that CA is a cover of XA implies that SA ⊆ e−1(XA) ⊆ e−1

A (⋃CA).

Hence qSA(

⋂G) ∩ SA = ∅, that is, (

⋂G) ∩ (qS

A)−1(SA) = ∅. In particular (⋂G) ∩ Z = ∅. Therefore

(⋂φ(G)) ∩ φ(Z) = ∅. �

7. Some consequences of Theorem 6.1

Since the identity function is a condensation and every base is a π-base, we have:

Theorem 7.1. Let C be a Gδ-dense subspace of XY . If X is a non compact locally compact secondcountable space, then the space C is not weakly pseudocompact.

Our first corollary includes Theorem 1.3.

Corollary 7.2. Let G be a metrizable separable locally compact non compact topological group and let κbe a cardinal number. Then, the space Cp(X,Gκ) is never weakly pseudocompact.

Proof. Assume that there exists a space X such that Cp(X,Gκ) is weakly pseudocompact. The spaceCp(X,Gκ) is homeomorphic to Cp(⊕ξ<κXξ, G) where each Xξ = X. It was proved in [9, Corollary6.5] that this implies that Cp(⊕ξ<κXξ, G) is Gδ-dense in G⊕ξ<κXξ . Because of Theorem 6.1, the spaceCp(X,Gκ) is not weakly pseudocompact; a contradiction. �

Corollary 7.3. For every Tychonoff space X and every cardinal number κ, the space Cp(X,Rκ) is notweakly pseudocompact.

Corollary 7.4. For every zero-dimensional space X and every cardinal number κ, the space Cp(X,Zκ)is not weakly pseudocompact.

Since the space of irrational numbers with its Euclidean topology P is homeomorphic to Zω, we obtain:

8 F. HERNANDEZ-HERNANDEZ, R. ROJAS-HERNANDEZ, AND A. TAMARIZ-MASCARUA

Corollary 7.5. For every zero-dimensional space X and every cardinal number κ, the space Cp(X,Pκ)is not weakly pseudocompact.

Corollary 7.6. The spaces Rκ, Nκ and Pκ are not weakly pseudocompact, for any cardinal κ.

The previous result gives us examples of non Lindelof R-factorizable Sanchez-Okunev countably com-pact Oxtoby countably compact topological groups with weight κ · ω, for any cardinal κ, which are notweakly pseudocompact. The space Rκ is even Telgarsky countably compact (see definitions in Section 3in [4]; see also Theorem 4.6 and Question 11.5 in the same article).

For a filter F on κ, a topological space X, and o ∈ X, we consider the space ΣF,oXκ := {x ∈ Xκ :

{ξ < κ : x(ξ) = o} ∈ F} which is the so called ΣF -product on Xκ based on o. If F is ω-complete (thatis, if for every sequence F0, ..., Fn, ... of elements in F ,

⋂n<ω Fn ∈ F), then ΣFXκ is Gδ-dense in Xκ.

Corollary 7.7. Let F be an ω-complete filter on κ. Let G be a metrizable separable locally compact noncompact topological group. Then ΣF,oG

κ is not weakly pseudocompact.

Let κ be a cardinal number such that cof(κ) > ω. Let Fω be equal to the filter on κ of all the subsetsof κ with countable complement. Then Fω is ω-complete. So:

Corollary 7.8. Let κ be a cardinal number such that cof(κ) > ω. Let G be a metrizable separable locallycompact non compact topological group. Then ΣFω,oG

κ is not weakly pseudocompact. In particular, theclassic Σ-products ΣRκ and ΣZκ are not weakly pseudocompact.

Since the Sorgenfrey line S can be condensed on the real line and the collection of open intervals area π-base in S, we conclude from Theorem 6.1 the following:

Corollary 7.9. For every cardinal number κ, Sκ and ΣSκ are not weakly pseudocompact.

We finish our list of corollaries by mentioning the Niemyntzki Plane M (see [7]). Observe that M isa Baire non Lindelof space; and in addition M can be condensed into the Euclidean semiplane. So, weobtain:

Corollary 7.10. For every cardinal number κ, the spaces Mκ and ΣMκ are not weakly pseudocompact.

In order to present our last remark we need the following definition. Let Y be a topological space. Aspace X is uY -discrete if every countable subset N of X satisfies: (i) N is closed, and (ii) every continuousfunction f : N → Y can be extended to a continuous function F : X → Y .

Remark 7.11. None of the hypotheses in Theorem 6.1 are essential as Corollary 7.5 shows when κ > c.On the other hand, separability is essential in Corollary 7.2. Indeed, in [2] it was proved that for azero-dimensional uZ-discrete space X and for the discrete space of cardinality κ > ω, D(κ), Cp(X,D(κ))is a weakly pseudocompact topological group (and non locally pseudocompact if X is infinite). (A moregeneral result is: for every weakly pseudocompact space Y and every uY -discrete space X, Cp(X,Y ) isweakly pseudocompact).

8. Open problems

Problem 8.1. Is it true that Cp(X,Y ) is not weakly pseudocompact when Y is metrizable separablelocally compact non compact? That is, is the version of Corollary 7.2 in which we do not ask the grouptopology structure on G true?

We know that for every uY -discrete space X (see definition before Remark 7.11), Cp(X,Y ) is weaklypseudocompact if Y is weakly pseudocompact (see [3]); then the following question arises:

Problem 8.2. Let X be a uY -discrete space. Must Y be weakly pseudocompact if Cp(X,Y ) is weaklypseudocompact?

References

[1] A. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics,Editor: J. van Mill, Atlantis Press/World Scientific, 2008.

[2] A. Dorantes-Aldama, R. Rojas-Hernandez, A. Tamariz-Mascarua, The partially pre-order set of compactifications ofCp(X, Y ), Topol. Algebra Applications, 3 (2015), 11-25.

[3] A. Dorantes-Aldama, R. Rojas-Hernandez, A. Tamariz-Mascarua, Weak pseudocompactness of spaces of continuousfunctions, Topology and its Appl., 196 (2015), 72-91.

[4] A. Dorantes-Aldama and D. Shakhmatov, Completeness and compactness properties in metric spaces, topologicalgroups and function spaces, manuscript.

Cp(X) IS NEVER WEAKLY PSEUDOCOMPACT 9

[5] A. Dorantes-Aldama and A. Tamariz-Mascarua, Some results on weakly pseudocompact spaces, Houston Journal.

[6] F. W. Eckertson, Sums, products, and mappings of weakly pseudocompact spaces, Topology and its Applications 72(1996) 149-157.

[7] R. Engelking. General Topology, Heldermann Verlag, Berlin 1989.[8] S. Garcıa-Ferreira and A. Garcıa-Maynez, On weakly-pseudocompact spaces, Houston Journal of Mathematics. 20 (1)

(1994) 145-159.

[9] S. Garcıa-Ferreira, R. Rojas-Hernandez, A. Tamariz-Mascarua, Completeness type properties on Cp(X, Y ) spaces,manuscript.

Facultad de Ciencias Fısico Matematicas, Universidad Michoacana de San Nicolas de Hidalgo, Morelia,Michoacan, Mexico

E-mail address: fhernandez@fismat.umich.

Centro de Ciencias Matematicas, UNAM, Campus Morelia, C.P. 58190, Morelia, Michoacan, MexicoE-mail address: satzchen@yahoo.com.mx

Departamento de Matematicas, Facultad de Ciencias, UNAM, Circuito exterior s/n, Ciudad Universitaria,CP 04510, Mexico D. F., Mexico

E-mail address: atamariz@unam.mx