TkachukAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 10, No. 1, 2009

pp. 39-48

Condensations of Cp(X) onto σ-compact spaces

V. V. Tkachuk
∗

Abstract. We show, in particular, that if nw(Nt) ≤ κ for any t ∈ T
and C is a dense subspace of the product

∏
{Nt : t ∈ T } then, for

any continuous (not necessarily surjective) map ϕ : C → K of C into a
compact space K with t(K) ≤ κ, we have Ψ(ϕ(C)) ≤ κ. This result has
several applications in Cp-theory. We prove, among other things, that
if K is a non-metrizable Corson compact space then Cp(K) cannot
be condensed onto a σ-compact space. This answers two questions
published by Arhangel’skii and Pavlov.

2000 AMS Classification: Primary 54H11, 54C10, 22A05, 54D06; Secondary
54D25, 54C25

Keywords: condensation, continuous image, Lindelöf Σ-space, σ-compact
space, topology of pointwise convergence, network weight, tightness, Lindelöf
space.

1. Introduction.

A weaker topology on a space X can be considered an approximation of the
topology of X. If this approximation has some nice properties then we can
obtain a lot of useful information about the space X. Thus it is natural to find
out when a space has a weaker compact topology. This is an old topic and an
extensive research has been done here both in general topology and descriptive
set theory.

The quest for nice condensations of function spaces had its origin in func-
tional analysis after Banach asked whether every separable Banach space has
a weaker compact metrizable topology. This problem was solved positively by
Pytkeev [9]. Answering a question of Arhangel’skii, Casarrubias–Segura showed
in [5] that function spaces of Cantor cubes have a weaker Lindelöf topology but
it is consistent that some of them do not have a weaker compact topology.

∗Research supported by Consejo Nacional de Ciencia y Tecnoloǵıa (CONACYT) de
México, grant 400200-5-38164-E



40 V. V. Tkachuk

Arhangel’skii and Pavlov [4] studied systematically when Cp(X) has a weaker
compact topology and formulated some open questions on weaker σ-compact
topologies on Cp(X). It is also worth mentioning that Marciszewski [7] gave a
consistent example of a space X ⊂ R such that Cp(X) does not have a weaker
σ-compact topology.

In this paper we consider product spaces N =
∏

t∈T Nt such that nw(Nt) ≤
κ for all t ∈ T . We prove that if C is a dense subspace of N and ϕ : C → K
is a continuous (not necessarily surjective) map of C into a compact space K
with t(K) ≤ κ then Ψ(ϕ(C)) ≤ κ, i.e., every closed subset of ϕ(C) is the
intersection of at most κ-many open subsets of ϕ(C). This result has several
important applications in Cp-theory. We establish, in particular, that if X
is an ω-monolithic space such that l(Cp(X)) = t(Cp(X)) = ω and Cp(X)
condenses onto a σ-compact space then X is cosmic. As a consequence, if
X is a non-metrizable Corson compact space then Cp(X) does not condense
onto a σ-compact space. This answers Questions 29 and 30 of the paper of
Arhangel’skii and Pavlov [4].

Any compact space of countable tightness has countable π-character (see [1,
Theorem 2.2.20]). This easily implies that if Cp(X) embeds in such a compact
space then X is countable. Therefore it is natural to conjecture that every
continuous image of Cp(X) has a countable network whenever it embeds in
a compact space of countable tightness. Another reason to believe that this
conjecture might be true is a theorem of Tkachenko [12] which states that if
a compact space K of countable tightness is a continuous image of a Lindelöf
Σ-group then K is metrizable. At the present moment nothing contradicts the
hypothesis that if G is a Lindelöf Σ-group and ϕ : G → K is a continuous map,
where K is compact and t(K) ≤ ω then ϕ(G) has a countable network. We
prove this conjecture for the spaces Cp(X) with the Lindelöf Σ-property.

2. Notation and terminology.

All spaces under consideration are assumed to be Tychonoff. If X is a space
then τ(X) is its topology and τ∗(X) = τ(X) \ {∅}; given an arbitrary set
A ⊂ X let τ(A,X) = {U ∈ τ(X) : A ⊂ U}. If x ∈ X then we write τ(x,X)
instead of τ({x},X). The space R is the real line with its natural topology and
N = ω \ {0}. If X and Y are spaces then Cp(X,Y ) is the space of real-valued
continuous functions from X to Y endowed with the topology of pointwise
convergence. We write Cp(X) instead of Cp(X, R). The expression X ≃ Y
says that the spaces X and Y are homeomorphic.

A family N of subsets of a space Z is called a network if for any U ∈ τ(Z)
there is N ′ ⊂ N such that

⋃
N ′ = U. The network weight nw(Z) of a space

Z is the minimal cardinality of a network in Z. A space X is called cosmic if
the network weight of X is countable. If x ∈ X then a family A is a network
of X at the point x if x ∈

⋂
A and for any U ∈ τ(x,X) there is A ∈ A such

that A ⊂ U.
A family B ⊂ τ∗(X) is a π-base of X at a point x ∈ X if for any U ∈ τ(x,X)

there is B ∈ B with B ⊂ U. The minimal cardinality of a π-base of X at x is



Condensations of Cp(X) onto σ-compact spaces 41

denoted by πχ(x,X) and πχ(X) = sup{πχ(x,X) : x ∈ X}. If ϕ is a cardinal
invariant then hϕ(X) = sup{ϕ(Y ) : Y ⊂ X} is the hereditary version of ϕ. If X
is a space and F is a closed subset of X then pseudocharacter ψ(F,X) of the set
F in the space X is the minimal cardinality of a family U ⊂ τ(F,X) such that⋂
U = F ; let ψ(X) = sup{ψ({x},X) : x ∈ X} and Ψ(X) = sup{ψ(F,X) : F is

a closed subset of X}.
The tightness t(X) of a space X is the minimal cardinal κ such that, for any

A ⊂ X, if x ∈ A then there is B ⊂ A with |B| ≤ κ such that x ∈ B. We use
the Russian term condensation to denote a continuous bijection. A space Z is
called κ-monolithic if for any A ⊂ Z with |A| ≤ κ, we have nw(A) ≤ κ.

If we have a product Z =
∏

t∈T Zt and A ⊂ T then ZA =
∏

t∈A Zt is
the A-face of Z and πA : Z → ZA is the natural projection. A set F ⊂ Z
depends on A ⊂ T if π−1

A
πA(F) = F ; if F depends on a set of cardinality ≤ κ

then we say that F depends on at most κ-many coordinates. A set E ⊂ Z
covers a face ZA if πA(E) = ZA. Suppose that, for every t ∈ T we have a
family Nt of subsets of Zt and let N = {Nt : t ∈ T}. If we have a faithfully
indexed set A = {t1, . . . , tn} ⊂ T and Ni ∈ Nti for each t ≤ n then let
[t1, . . . , tn,N1, . . . ,Nn] = {x ∈ Z : x(ti) ∈ Ni for all i = 1, . . . ,n}. A set H ⊂ Z
is called N -standard (or standard if N is clear) if H = [t1, . . . , tn,N1, . . . ,Nn]
for some t1, . . . , tn ∈ T and Ni ∈ Nti for all i ≤ n. In this case we let
supp(H) = A and r(H) = n. We also consider that H = Z is the unique
standard subset of Z such that r(H) = 0. Given any point x ∈ Z and A ⊂ T
the set 〈x,A〉 = {y ∈ Z : y(t) = x(t) for any t ∈ A} is closed in Z. If A ⊂ T
then the face ZA is called κ-residual if |T \A| ≤ κ. Say that a non-empty closed
set F ⊂ K is κ-large if, for any x ∈ F and any finite A ⊂ T , the set 〈x,A〉 ∩ F
covers a κ-residual face of K.

All other notions are standard and can be found in [6] and [3].

3. Nice continuous images of function spaces.

Our results will be obtained by strengthening a result of Shirokov [11]. Al-
though our modifications of Shirokov’s method are minimal, we give complete
proofs because the paper [11] has never been translated and, even in Russian,
it is completely out of access for a Western reader. In particular, we present
the proof of the following lemma established in [11].

Lemma 3.1. Given an infinite cardinal κ suppose that nw(Nt) ≤ κ for any
t ∈ T and N =

∏
t∈T Nt. Assume that C ⊂ N is dense in N, and we have a

compact extension Kt of the space Nt for any t ∈ T. If a set F ⊂ K =
∏

t∈T Kt
is κ-large then there exists a Gκ-set G in the space K such that F ⊂ G and
F ∩ C = G ∩ C. In particular, F ∩ C is a Gκ-subset of C.

Proof. We can assume, without loss of generality, that K \ F 6= ∅. For every
t ∈ T fix a network Nt in the space Nt such that |Nt| ≤ κ; we will need the
family Mt = {clKt (N) : N ∈ Nt}. If M = {Mt : t ∈ T} then the M-standard
subsets of K will be called standard. It is easy to see that

(1) the family H of all standard subsets of K is a network in K at every x ∈ C.



42 V. V. Tkachuk

Given standard sets P and P ′ say that P ′ � P if P = [t1, . . . , tn,M1, . . . ,Mn]
and there exists a natural k ≤ n such that P ′ = [ti1, . . . , tik,Mi1, . . . ,Mik ] for
some distinct i1, . . . , ik ∈ {1, . . . ,n}; if k < n then we write P

′ ≺ P . We also
include here the case when k = 0 so P ′ = K � P for any standard set P .
Say that a standard set P is minimal if P ∩ F = ∅ but P ′ ∩ F 6= ∅ whenever
P ′ ≺ P . It follows from (1) that

(2) for any x ∈ C \ F there exists a minimal standard set P such that x ∈ P .

It will be easy to finish our proof if we establish that

(3) the family S of minimal standard sets has cardinality not exceeding κ.

Assume, toward a contradiction that |S| > κ. Then we can choose S0 ⊂ S
such that |S0| = κ

+ and there exists n ∈ ω with r(P) = n for all P ∈ S0.
Observe first that

(4) if A ⊂ T , a set D ⊂ K covers the face KT \A and a standard set P is disjoint
from D then supp(P) ∩ A 6= ∅.

Indeed, if supp(P) = {t1, . . . , tk} ⊂ T \ A and P = [t1, . . . , tk,M1, . . . ,Mk]
then it follows from πT \A(D) = KT \A that there exists a point x ∈ D such
that x(ti) ∈ Mi for all i ≤ k. Therefore x ∈ D ∩ P which is a contradiction.

The set F being κ-large, there exists A1 ⊂ T with |A1| ≤ κ such that F
covers the face KT \A1 . The property (4) shows that supp(P) ∩A1 6= ∅ for any
P ∈ S0. There exists a point t1 ∈ A1 such that the family S

′
0 = {P ∈ S0 : t1 ∈

P} has cardinality κ+. Since |Mt1| ≤ κ, we can find a family S1 ⊂ S
′
0 and

M1 ∈ Mt1 such that |S1| = κ
+ and [t1,M1] � P for any P ∈ S1.

Proceeding by induction assume that k < n and we have a set Ak ⊂ T with
|Ak| ≤ κ and a family Sk such that |Sk| = κ

+ and, for some t1, . . . , tk ∈ Ak and
Mi ∈ Mti (i = 1, . . . ,k), we have [t1, . . . , tk,M1, . . . ,Mk] � P for every P ∈
Sk. Therefore P = [t1, . . . , tk,s1, . . . ,sn−k,M1, . . . ,Mk,E1, . . . ,En−k] for every
P ∈ Sk; let Q(P) be the set in which s1 and E1 are omitted from the definition
of P , i.e., Q(P) = [t1, . . . , tk,s2, . . . ,sn−k,M1, . . . ,Mk,E2, . . . ,En−k]. It is
clear that Q(P) ≺ P ; since P is minimal, the set Q(P) intersects F for each
P ∈ Sk.

Fix a set R ∈ Sk and let F
′ = F ∩ Q(R). The set F being κ-large, we can

find A ⊂ T with |A| ≤ κ such that F ′ covers the face KT \A and hence the face
KT \(A∪Ak) as well. Let Ak+1 = A∪Ak and observe that every set P ∈ Sk is dis-
joint from F ′; this, together with (4) shows that supp(P)∩Ak+1 6= ∅. Suppose
for a moment that P = [t1, . . . , tk,s1, . . . ,sn−k,M1, . . . ,Mk,E1, . . . ,En−k] ∈
Sk and {s1, . . . ,sn−k} ∩ Ak+1 = ∅. Since F

′ covers the face KT \Ak+1 , we can
find a point x ∈ F ′ such that x(si) ∈ Ei for all i ≤ n − k; since also x(ti) ∈ Mi
for all i ≤ k because x ∈ Q(R), we conclude that x ∈ F ′∩P . This contradiction
implies that {s1, . . . ,sn−k}∩Ak+1 6= ∅ and hence the set supp(P)\{t1, . . . , tk}
intersects the set Ak+1 for any P ∈ Sk \ {R}.

Therefore we can choose a family Sk+1 ⊂ Sk of cardinality κ
+ together with

a point tk+1 ∈ Ak+1 \ {t1, . . . , tk} and a set Mk+1 ∈ Mtk+1 such that we have
[t1, . . . , tk+1,M1, . . . ,Mk+1] � P for any P ∈ Sk+1. As a consequence, our
inductive procedure can be continued to construct a family Sn ⊂ S such that



Condensations of Cp(X) onto σ-compact spaces 43

|Sn| = κ
+ while [t1, . . . , tn,M1, . . . ,Mn] � P for any P ∈ Sn. Recalling that

r(P) = n, we conclude that we have the equality P = [t1, . . . , tn,M1, . . . ,Mn]
for each P ∈ Sn; this contradiction shows that |S| ≤ κ, i.e., (3) is proved.

It is straightforward that G = K \ (
⋃
S) is a Gκ-subset of K such that

F ⊂ G and F ∩ C = G ∩ C. �

The following result generalizes Theorem 1 of [11].

Theorem 3.2. Given an infinite cardinal κ suppose that nw(Nt) ≤ κ for any
t ∈ T and C ⊂ N =

∏
t∈T Nt is a dense subspace of N. Assume additionally

that we have a continuous (not necessarily surjective) map ϕ : C → L of C into
a compact space L. If y ∈ C′ = ϕ(C) and hπχ(y,L) ≤ κ then ψ(y,C′) ≤ κ.

Proof. There is no loss of generality to assume that C′ is dense in L. Choose
a compact extension Kt of the space Nt for any t ∈ T ; then K =

∏
t∈T Kt is a

compact extension of both N and C. There exist continuous maps Φ : βC → L
and ξ : βC → K such that Φ|C = ϕ and ξ(x) = x for any x ∈ C. It is clear
that both Φ and ξ are surjective.

For every t ∈ T fix a network Nt in the space Nt such that |Nt| ≤ κ and
let Mt = {clKt (N) : N ∈ Nt}. If M = {Mt : t ∈ T} then the M-standard
subsets of K will be called standard. Our first step is to prove that

(5) the set Fy = ξ(Φ
−1(y)) is κ-large.

Fix a point x ∈ Fy , a finite A ⊂ T and consider the set P = 〈x,A〉 =
{x′ ∈ K : x′(t) = x(t) for all t ∈ A}. It follows from P ∩ Fy 6= ∅ that
ξ−1(P) ∩ Φ−1(y) 6= ∅ and hence y ∈ Q = Φ(ξ−1(P)). The set Q is compact
and it follows from hπχ(y,L) ≤ κ that we can choose a π-base B of the space
Q at the point y such that |B| ≤ κ. For every B ∈ B pick a set OB ∈ τ(L) such
that ∅ 6= OB ∩ Q ⊂ OB ∩ Q ⊂ B. It follows in a standard way from c(K) ≤ κ
that

(6) for any U ∈ τ∗(L), the set clK (ϕ
−1(U)) depends on at most κ-many coor-

dinates and coincides with the set ξ(clβC (Φ
−1(U))).

Apply (6) to find a set S ⊂ T of cardinality at most κ for which A ⊂ S
and the set DB = ξ(clβC (Φ

−1(OB ))) depends on S for any B ∈ B. The face
KT \S is residual; to show that P ∩ Fy covers KT \S fix any point w ∈ KT \S
and consider the set E = {z ∈ K : πT \S (z) = w and πS (z) ∈ πS (P)}. Clearly,
E is a non-empty compact subset of P .

Fix any B ∈ B; it follows from OB ∩Q 6= ∅ that there is a point u ∈ ξ
−1(P)

such that Φ(u) ∈ OB; thus u ∈ Φ
−1(OB ) which shows that ξ(u) ∈ DB ∩ P .

Define a point u′ ∈ K by the equalities πT \S (u
′) = w and πS (u

′) = πS (ξ(u)).
Since the sets DB and P depend on S, we conclude that u

′ ∈ DB ∩ P . On the
other hand, πS (u

′) ∈ πS (P) so u
′ ∈ E, and therefore E ∩ DB 6= ∅.

As a consequence, Φ(ξ−1(E)) ∩ OB 6= ∅ and hence Φ(ξ
−1(E)) ∩ B 6= ∅ for

any B ∈ B; since Φ(ξ−1(E)) is a closed subset of Q and B is a π-base of Q
at y, we must have y ∈ Φ(ξ−1(E)) which implies that ξ−1(E) ∩ Φ−1(y) 6= ∅
and hence E ∩ Fy 6= ∅. If v ∈ E ∩ Fy then w = πT \S (v) ∈ πT \S (P ∩ Fy ); the



44 V. V. Tkachuk

point w ∈ KT \S was chosen arbitrarily so P ∩Fy covers KT \S and hence (5) is
proved.

By Lemma 3.1 there exists a Gκ-set G in the space K such that Fy ⊂ G and
G ∩ C = Fy ∩ C = ϕ

−1(y). Therefore we can choose a family F of compact
subsets of K such that |F| ≤ κ and C \ Fy ⊂

⋃
F ⊂ K \ Fy . For any F ∈ F

the set WF = L \ Φ(ξ
−1(F)) is an open neighbourhood of y in L and it is

straightforward that H =
⋂
{WF : F ∈ F} is a Gκ-subset of L such that

H ∩ C′ = {y}. �

Corollary 3.3. Suppose that C is a dense subspace of a product N =
∏

t∈T Nt
such that nw(Nt) ≤ κ for each t ∈ T. Assume that K is a compact space with
t(K) ≤ κ and ϕ : C → K is a continuous (not necessarily surjective) map; let
C′ = ϕ(C). Then every closed subspace of C′ is a Gκ-set, i.e., Ψ(C

′) ≤ κ; in
particular, ψ(C′) ≤ κ.

Proof. Fix a non-empty closed set F ′ in the space C′ and let F = clK (F
′).

Consider the quotient map p : K → KF obtained by contracting the set F
to a point and let q = p|C′. It is easy to see that we have the inequalities
t(KF ) ≤ t(K) ≤ κ; denote by y the point of the space KF represented by F
and let C′′ = p(C′). It follows from [1, Theorem 2.2.20] that hπχ(y,KF ) ≤ κ
so Theorem 3.2, applied to the map p ◦ ϕ, implies that ψ(y,C′′) ≤ κ. Since
F ′ = q−1(y), we conclude that F ′ is a Gκ-subset of C

′. �

Corollary 3.4. Suppose that C is a dense subspace of a product N =
∏

t∈T Nt
such that nw(Nt) ≤ κ for each t ∈ T. Assume additionally that l(C) ≤ κ and
K is a compact space with t(K) ≤ κ such that there exists a continuous (not
necessarily surjective) map ϕ : C → K. If C′ = ϕ(C) then hl(C′) ≤ κ.

Proof. We have l(C′) ≤ κ while every closed subspace of the space C′ is a
Gκ-set by Corollary 3.3. Now, a standard proof shows that hl(C

′) ≤ κ. �

Corollary 3.5. If C is a dense subspace of a product of cosmic spaces and
K is a compact space then, for any continuous map ϕ : C → K, we have
Ψ(ϕ(C)) ≤ t(K).

The last corollary has several applications in Cp-theory. Let us start with
the following observation.

Proposition 3.6. (Folklore). If the space of a topological group G embeds in
a compact space of countable tightness then G is metrizable. In particular, if
Cp(X) embeds in a compact space of countable tightness then Cp(X) is second
countable and hence X is countable.

Proof. Assume that G is a dense subspace of a compact space K with t(K) ≤ ω.
Then πχ(g,G) = πχ(g,K) ≤ ω (see [1, Theorem 2.2.20]) and hence we have
the equality χ(g,G) = πχ(g,G) = ω for any g ∈ G (see [2, Proposition 1.1]) so
G is metrizable. �

The following result is a curious generalization of Proposition 3.6 for the
case of condensations.



Condensations of Cp(X) onto σ-compact spaces 45

Corollary 3.7. For any X, the space Cp(X) condenses onto a space embed-
dable in a compact space of countable tightness if and only if Cp(X) condenses
onto a second countable space.

Proof. Apply Corollary 3.5 and the equality ψ(Cp(X)) = iw(Cp(X)). �

However, it would be interesting to find out whether any continuous image of
Cp(X) embeddable in a compact space of countable tightness has to be cosmic
or even metrizable. It follows from Corollary 3.3 that such an image is a perfect
space. The following theorem shows that this conjecture is true when Cp(X)
is a Lindelöf Σ-space.

Theorem 3.8. Suppose that ϕ : Cp(X) → K is a continuous (not necessarily
surjective) map and K is a compact space with t(K) ≤ ω; let Y = ϕ(Cp(X)).
Then

(i) Y is a perfect space of countable π-weight;
(ii) if Cp(X) is a Lindelöf Σ-space then Y is cosmic.

Proof. That Y is perfect is an immediate consequence of Corollary 3.3. Since
ω1 is a precaliber of Cp(X), it has to be also a precaliber of Y and hence

of Y . The space Y being compact, the cardinal ω1 is a caliber Y ; it follows
from t(Y ) ≤ ω that Y has a point-countable π-base [10]. This implies that
πw(Y ) = ω and hence πw(Y ) = ω as well, i.e., we settled (i).

If Cp(X) is a Lindelöf Σ-space then Cp(X) × Cp(X) is Lindelöf. The space
Y × Y is a continuous image of Cp(X) × Cp(X) ≃ Cp(X ⊕ X) so we can apply
Corollary 3.4 to convince ourselves that Y ×Y is hereditarily Lindelöf and hence
Y condenses onto a second countable space. This, together with the Lindelöf
Σ-property of Y implies that nw(Y ) ≤ ω and hence (ii) is proved. �

In the sequel we will need the following lemma from [13].

Lemma 3.9. If Cp(X) =
⋃

n∈ω Fn and every Fn is closed in Cp(X) then there
exists n ∈ ω such that Cp(X) embeds in Fn.

Theorem 3.10. Suppose that l(Xn) = ω for all n ∈ N and Cp(X) is Lindelöf.
If Cp(X) condenses onto a σ-compact space Y then the space X is separable
and ψ(Y ) = ω.

Proof. Fix a condensation ϕ : Cp(X) → Y and a family {Kn : n ∈ ω} of
compact subsets of Y such that Y =

⋃
n∈ω Kn. The set Fn = ϕ

−1(Kn) is
closed in Cp(X) for every n ∈ ω. If n ∈ ω and S is an uncountable free
sequence in Kn then S

′ = ϕ−1(S) is an uncountable free sequence in Fn which
is impossible because l(Fn) ≤ l(Cp(X)) = ω and t(Fn) ≤ t(Cp(X)) = ω. This
contradiction shows that Kn has no uncountable free sequences and therefore
t(Fn) ≤ ω for any n ∈ ω.

Apply Lemma 3.9 to see that there exists n ∈ ω such that C ≃ Cp(X) for
some C ⊂ Fn. Since ϕ|C maps C into Kn, Corollary 3.5 shows that ψ(ϕ(C)) ≤
ω. Since ϕ|C is a condensation, we have ψ(Cp(X)) = ψ(C) ≤ ψ(ϕ(C)) = ω
and hence d(X) = ψ(Cp(X)) = ω, i.e., X is separable as promised.



46 V. V. Tkachuk

It follows from ψ(Cp(X)) = ω that Cp(X) \ {f} is an Fσ-set for any f ∈
Cp(X). The space Cp(X) being Lindelöf, Cp(X) \ {f} is Lindelöf as well.
Therefore Y \ {y} is Lindelöf for any y ∈ Y ; this implies that ψ(Y ) ≤ ω. �

Corollary 3.11. Suppose that X is an ω-monolithic space such that Cp(X)
is Lindelöf and Xn is Lindelöf for any n ∈ N. If Cp(X) condenses onto a
σ-compact space Y then nw(X) = nw(Y ) = ω.

Proof. Theorem 3.10 shows that the space X must be separable so nw(X) = ω
by ω-monolithity of X. Therefore nw(Y ) ≤ nw(Cp(X)) = nw(X) = ω. �

The following result gives a complete answer (in a much stronger form) to
Problems 29 and 30 from the paper [4].

Corollary 3.12. If X is an ω-monolithic compact space such that Cp(X) is
Lindelöf and can be condensed onto a σ-compact space then X is metrizable.
In particular, if X is a non-metrizable Corson compact space then Cp(X) does
not condense onto a σ-compact space.

Corollary 3.13. Under MA+¬CH if K is a compact space such that Cp(K)
is Lindelöf and can be condensed onto a σ-compact space then X is metrizable.

Proof. It is a result of Reznichenko (see [3, Theorem IV.8.7]) that MA+¬CH
together with the Lindelöf property Cp(K) implies that K is ω-monolithic so
we can apply Corollary 3.12 to see that K is metrizable. �

Corollary 3.14. Assume that Cp(X) is a Lindelöf Σ-space and there exists a
condensation of Cp(X) onto a σ-compact space Y . Then nw(X) = nw(Y ) = ω.

Proof. Denote by υX the Hewitt realcompactification of X. It is evident that
Cp(X) is a continuous image of the space Cp(υX). Besides, Z = υX is a
Lindelöf Σ-space by [8, Corollary 3.6] ; since Cp(Z) is also a Lindelöf Σ-space
(see [14, Theorem 2.3]), Corollary 3.11 is applicable to Z and we can conclude
that nw(Z) = nw(Y ) = ω. Since X ⊂ Z, we have nw(X) ≤ nw(Z) = ω. �

4. Open problems.

There are still many opportunities for discovering interesting facts about
condensations of function spaces. The list below shows some possible lines of
research in this direction.

Problem 4.1. Suppose that K is a compact space of countable tightness and
ϕ : Cp(X) → K is a continuous map. Is it true that ϕ(Cp(X)) is cosmic or
even metrizable?

Problem 4.2. Suppose that Cp(X) is Lindelöf, K is a compact space of count-
able tightness and ϕ : Cp(X) → K is a continuous map. Is it true that
ϕ(Cp(X)) is cosmic or even metrizable?



Condensations of Cp(X) onto σ-compact spaces 47

Problem 4.3. Suppose that Cp(X) is hereditarily Lindelöf, K is a compact
space of countable tightness and ϕ : Cp(X) → K is a continuous map. Is it
true that ϕ(Cp(X)) is cosmic or even metrizable?

Problem 4.4. Suppose that X is compact, K is a compact space of countable
tightness and ϕ : Cp(X) → K is a continuous map. Is it true that ϕ(Cp(X))
is cosmic or even metrizable?

Problem 4.5. Is it true that, for any cardinal κ and any compact space K
with t(K) ≤ ω, if ϕ : Rκ → K is a continuous map then ϕ(Rκ) is cosmic or
even metrizable?

Problem 4.6. Suppose that K is a compact space of countable tightness, G is
a topological group with the Lindelöf Σ-property and ϕ : G → K is a continuous
map. Is it true that ϕ(G) is cosmic?

Problem 4.7. Suppose that Cp(X) is Lindelöf and there exists a condensation
of Cp(X) onto a σ-compact space Y . Must Y be cosmic?

Problem 4.8. Suppose that Cp(X) is Lindelöf and there exists a condensation
of Cp(X) onto a σ-compact space Y . Must X be separable?

Problem 4.9. Suppose that Cp(X) condenses onto a space of countable π-
weight. Must X be separable?

Problem 4.10. Suppose that Cp(X) is Lindelöf and ϕ : Cp(X) → Y is a
continuous onto map. Is it true that every compact subspace of Y has countable
tightness?

Problem 4.11. Suppose that K is Eberlein compact and ϕ : Cp(K) → Y is
a continuous surjective map of Cp(X) onto a σ-compact space Y . Must Y be
cosmic?

Problem 4.12. Suppose that K is Corson compact and ϕ : Cp(K) → Y is
a continuous surjective map of Cp(X) onto a σ-compact space Y . Must Y be
cosmic?

Problem 4.13. Suppose that X is a space such that Cp(X) has the Lindelöf
Σ-property and ϕ : Cp(K) → Y is a continuous surjective map of Cp(X) onto
a σ-compact space Y . Must Y be cosmic?

Problem 4.14. Suppose that Cp(X) is Lindelöf and X
n is also Lindelöf for

any n ∈ N. Assume additionally that there exists a condensation of Cp(X)
onto a σ-compact space Y . Must Y be cosmic?

Problem 4.15. Suppose that K is a perfectly normal compact space. Is it true
that every σ-compact continuous image of Cp(X) has a countable network?



48 V. V. Tkachuk

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Received January 2008

Accepted January 2009

Vladimir V. Tkachuk (vova@xanum.uam.mx)
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av.
San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, C.P. 09340, México D.F.