The partially pre-order set of compactifications of Cp(X, Y)

Abstract

In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).
We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:
(a) Y is not locally compact,
(b) X has only one non isolated point and Y is not compact.
Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:
(i) X has a non-isolated point with countable character,
(ii) X is not pseudocompact,
(iii) X is infinite, pseudocompact and Cp(X) is normal,
(iv) X is an infinite generalized ordered space.
Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

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Topol. Algebra Appl. 2015; 3:11–25
Research Article Open Access
A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa*
The partially pre-ordered set of
compactications of Cp(X,Y)
DOI 10.1515/taa-2015-0002
Received September 19, 2014; revised January 16, 2015; accepted April 3, 2015
Abstract: In the set of compactications of Xwe consider the partial pre-order dened by (W,h)≤X(Z,g)
if there is a continuous function f:Z→W,such that (f∘g)(x) = h(x)for every x∈X. Two elements
(W,h)and (Z,g)of K(X)are equivalent, (W,h)≡X(Z,g), if there is a homeomorphism h:W→Zsuch
that (f∘g)(x) = h(x)for every x∈X. We denote by K(X)the upper semilattice of classes of equivalence of
compactications of Xdened by ≤Xand ≡X. We analyze in this article K(Cp(X,Y)) where Cp(X,Y)is the
space of continuous functions from Xto Ywith the topology inherited from the Tychono product space YX.
We write Cp(X)instead of Cp(X,R).
We prove that for a rst countable space Y,K(Cp(X,Y)) is not a lattice if any of the following cases happen:
(a) Yis not locally compact,
(b) Xhas only one non isolated point and Yis not compact.
Furthermore, K(Cp(X)) is not a lattice when Xsatises one of the following properties:
(i) Xhas a non-isolated point with countable character,
(ii) Xis not pseudocompact,
(iii) Xis innite, pseudocompact and Cp(X)is normal,
(iv) Xis an innite generalized ordered space.
Moreover, K(Cp(X)) is not a lattice when Xis an innite Corson compact space, and for every space X,
K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.
Keywords: Compactications of spaces of continuous functions, lattices, b-lattices, k-embedded subsets, C-
embedded subsets, retracts, weakly pseudocompact spaces
MSC: Primary 54C35, 54D35, 03G10, Secondary 54D45, 54C45.
1Introduction, basic denitions and notations
Throughout this article all topological spaces are considered Tychono and with more than one point if the
contrary is not specied. The symbol ωdenotes both the rst innite cardinal number and the natural num-
bers. For an ordinal number α,[0,α)denotes the space of ordinals strictly less than αwith its order topology,
and αdenotes the discrete space of cardinality |α|; in particular 2means the discrete space {0,1}. For a space
Xand a subset Aof X, clXA, or simply clA, if there is no possibility for confusion, is the closure of Ain X. A
subset Aof a space Xis a Gδsubset of Xif there is a countable collection {Un:n<ω}of open subsets of X
*Corresponding Author: Á. Tamariz-Mascarúa: Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exte-
rior s/n, Ciudad Universitaria, CP 04510, México D. F., México, E-mail: atamariz@unam.mx
A. Dorantes-Aldama: Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exterior s/n, Ciudad Universitaria,
CP 04510, México D. F., México, E-mail: alejandro_dorantes@hotmail.com
R. Rojas-Hernández: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma del Estado de México, Insti-
tuto Literario No. 100, Col. Centro, C.P. 50000, Toluca, Estado de México, México, E-mail: satzchen@yahoo.com.mx

12 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
such that A=n<ωUn. A subset Aof a space Xis Gδ-dense in Xif every non-empty Gδsubset of Xintersects
A.
Recall that a partial order (Z,≤)is a lattice if every pair of elements a,b∈Zhas a least upper bound and
a greatest lower bound. A lattice (Z,≤)is complete if every non-empty subset of Zhas a least upper bound
and a greatest lower bound.
A pair (Z,g)(respectively, (gX,g)) is a compactication of Xif Z(resp., gX) is a compact space and g:
X→Z(resp., g:X→gX) is an embedding such that g[X]is dense in Z(resp., in gX). Let (W,h)and (Z,g)
be two compactications of X. We say that (W,h)≤X(Z,g)(or just (W,h)≤(Z,g)), if there is no confusion
regarding the identity of the space Xcompactied by (W,h)and (Z,g)if there is a continuous function f:
Z→Wsuch that g=f∘h.These compactications (W,h)and (Z,g)are equivalent if (W,h)≤X(Z,g)and
(Z,g)≤X(W,h). We denote this fact by W≡XZ(or by W≡Zif there is no possibility for confusion). Observe
that two compactications (W,h)and (Z,g)of Xare equivalent if there is a homeomorphism f:Z→W
such that (h∘f)(x) = g(x)for every x∈X. We denote by K(X)the ordered set of ≡X-equivalence classes
of compactications of Xwith the partial order dened by ≤X. This partial order in K(X)will be denoted by
the same symbol ≤X. The partially ordered set (K(X),≤X)is a complete upper semilattice. The symbol (βX,β)
denotes the Stone-Čech compactication of X, and (αX,α)denotes the one-point compactication of Xfor a
locally compact space X.
A compactication (bX,b)of a space Xis primary if it is obtained by collapsing a compact subspace
of the Stone-Čech remainder βX \Xto a single point. (In some literature this class of compactications are
called simple. In this paper we follow the terminology used in [12]). A space Xis weakly pseudocompact if
Xis Gδ-dense in one of its compactications, and is primary pseudocompact if Xis Gδ-dense in a primary
compactication.
We say that (K(X),≤X)is a b-lattice if the primary compactications of Xare dense in (K(X),≤X); that is,
if for each (bX,b)∈K(X),there is a compact set K⊆β X \Xsuch that βX/K≤XbX.Observe that if Xis locally
compact then K(X)is a b-lattice because the one-point compactication of Xis a primary compactication.
We denote by Cp(X,Y)the space of continuous functions from Xto Ywith the topology of the pointwise
convergence which is that inherited from the product topology in YX. The product topology is generated by
the sets of the form:
[x1,. . . ,xn;B1,. . . ,Bn] = {f∈YX:f(xi)∈Bi,i= 1,. . . ,n}
where n<ω,{x1,. . . ,xn} ⊆ X,and B1,. . . ,Bnare open subsets of Y, as a base. So, the topology in
Cp(X,Y)is generated by the collection {[x1,. . . ,xn;B1,. . . ,Bn]∩Cp(X,Y) : n<ω,{x1,. . . ,xn} ⊆
X,and B1,. . . ,Bnare open subsets of Y}as a base. When Yis equal to the real line, we just write Cp(X)
instead of Cp(X,R).
In [5] some new results about weakly pseudocompact spaces were obtained, and the authors improved
several results on this topic obtained by F.W. Eckertson [6] and S. García-Ferreira and A. García-Máynez [8].
Part of the discussions made in [5] are related to the structure of the partially ordered set (K(X),≤X). In par-
ticular, in [5] the following results were proved:
Proposition 1.1. Let Xbe a space. If K(X)is a b-lattice, then Xis weakly pseudocompact if and only if Xis
primary pseudocompact.
Corollary 1.2. If Xis not locally pseudocompact and K(X)is a b-lattice, then Xis not weakly pseudocompact.
Theorem 1.3. Let Xbe a rst countable space. Then K(Xκ)is a b-lattice (resp., lattice, complete lattice) if and
only if Xis locally compact and either κis nite or Xis compact. In particular, K(ωκ)and K(Rκ)are b-lattices
(resp., lattice, complete lattice) i κ<ω, and K(Sκ)is not a b-lattice for every κ>0, where Sis the Sorgenfrey
line.
Proposition 1.4. Let Dbe a dense subspace of ωκ. Then, K(D)is b-lattice (resp., lattice, complete lattice) if
and only if κ<ω.

The partially pre-ordered set of compactications of Cp(X,Y)|13
The main goal in this article is to analyze (K(Cp(X,Y)),≤Cp(X,Y))where Yis a rst countable space. In partic-
ular, our concern will be centered in knowing when (K(Cp(X,Y)),≤Cp(X,Y))is a b-lattice.
The partially ordered set (K(X),≤X)has been studied in several papers (see for example [9] and [13]). In
this article we closely follow [14] and [15]. It is worth mentioning that two classic texts that develop the theory
of compactications of Hausdor spaces are [3] and [11].
In Section 2, we include some basic results about K(Cp(X,Y)) and we prove that if Yis rst countable
and Xis countable, then K(Cp(X,Y)) is a lattice if and only if Yis locally compact and either Xis nite or Y
is compact and Xis discrete. Sections 3 and 4 are devoted to introducing and analyzing k- and ˜
k-embedded
subspaces (Y⊆Xis k-embedded in Xif for every compactication bY of Y, there is a compactication aX of
Xsuch that claX Y≤b Y. A space Yis 
k-embedded in Xif Yis homeomorphic to a k-embedded subspace of X)
and we prove that all closed subsets of a locally compact space and all retracts are k-embedded subspaces. As
a consequence we obtain that for a rst countable non-locally compact space Y,K(Cp(X,Y)) is not a lattice,
no matter what the space Xis, and that K(Cp([0,α),Y)) and K(Cp(Ψ(A),Y)) are not lattices for every ordinal
number α>ωand every noncountable almost disjoint family A, where Yis rst countable. In Section 5 we
prove that K(Cp(X,Y)) is not a lattice when Yis a non-compact rst countable space and Xis either a non-
pseudocompact zero-dimensional space, or Xhas only one non isolated point. The main theorems appear in
Section 6: The semilattice K(Cp(X)) is not a lattice when Xhas at least one of the following properties: (i) X
has a non-isolated point with countable character, (ii) Xis not pseudocompact, (iii) Xis pseudocompact and
Cp(X)is normal, or (iv) Xis an innite generalized ordered space. Additionally, K(Cp(X)) is not a lattice when
Xis an innite Corson compact space, and for every space X,K(Cp(Cp(X))) is not a lattice. Finally, in Section
7 we list some unsolved problems.
2Some rst remarks about K(Cp(X,Y))
Lemma 2.1. [14] If K(X)is a b-lattice, then K(X)is a lattice.
Theorem 2.2. [15] If Xis a rst countable space, then Xis locally compact if and only if K(X)is a complete
lattice.
Corollary 2.3. If Xis a rst countable space then the following are equivalent:
1. Xis locally compact,
2. K(X)is a complete lattice,
3. K(X)is a lattice, and
4. K(X)is a b-lattice.
Related to Theorem 1.3, we have the following two propositions and one example. We will obtain a more
general result in Corollary 3.15 below.
Proposition 2.4. For a zero-dimensional space X,K(Cp(X,Z)κ)(respectively, K(C*
p(X,Z)κ)) is a b-lattice
(resp., lattice, complete lattice) if and only if Xand κare nite.
Proof. Indeed, on the one hand Cp(X,Z)κ(resp., C*
p(X,Z)κ) is a dense subset of Z|X|·κ. If either Xor κis
innite, [0,1]|X|·κis a connected compactication of Cp(X,Z)κ(resp., C*
p(X,Z)κ)). This last space is dense in
Z|X|·κ. So, K(Cp(X,Z)κ)(resp., K(C*
p(X,Z)κ)) is not a b-lattice (see Corollary 4.2 and Theorem 4.5 in [5]). On
the other hand, if |X|and κare nite, then Cp(X,Z)κ=Z|X|·κis locally compact and so, K(Cp(X,Z)κ)is a
b-lattice.
Example 2.5. Aσ-compact space Zfor which K(Z)is not a b-lattice: If Xis an innite Eberlein compact zero-
dimensional space, then Cp(X,Z)is σ-compact (see Theorem 4.18 in [4]) but K(Cp(X,Z)κ)is not a lattice for
every cardinal κ.

14 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
With similar arguments to those given in the proof of Proposition 2.4, we obtain:
Proposition 2.6. For every zero-dimensional space Xand every cardinal number κ,K(Cp(X,Q)κ)(resp.,
K(Cp(X,Q)κ),K(Cp(X,ωω))κ,K(C*
p(X,ωω))κ) is not a lattice.
If Xis a connected space and Yis a totally disconnected space such that K(Y)is a b-lattice, then, since
Cp(X,Y)∼
=Y, trivially K(Cp(X,Y)) is a b-lattice. So, in order to avoid trivial cases, and inspired in Proposi-
tions 2.4 and 2.6, we are tempted to request that Xand Ybe spaces that satisfy that Cp(X,Y)be dense in XY.
Nevertheless, this request will only be explicitly necessary in a few cases (see, for example Lemma 2.7 and
Theorem 2.8, below) as we will conrm throughout this article.
Lemma 2.7. Assume that Cp(X,Y)is dense in YX. Then, Cp(X,Y)is locally compact if and only if Yis locally
compact and either Xis nite or Yis compact and Xis discrete.
Proof. Assume that Cp(X,Y)is locally compact. Let y∈Y. Let Vbe an open set with y∈V. Since Cp(X,Y)
is dense in YX, we can take g∈[x;V]∩Cp(X,Y). Then, there exist n<ω,x1, ..., xn∈Xand open subsets of
Y,A1, ..., Ansuch that g∈[x1, ..., xn;A1, ..., An]∩Cp(X,Y)⊆clCp(X,Y)([x1, ..., xn;A1, ..., An]∩Cp(X,Y)) ⊆
[x;V]∩Cp(X,Y)where clCp(X,Y)([x1, ..., xn;A1, ..., An]∩Cp(X,Y)) is compact. Again, using the fact that
Cp(X,Y)is dense in YXwe can prove that
T=clCp(X,Y)([x0, ..., xk;A0, ..., Ak]∩Cp(X,Y))
is equal to
{h∈Cp(X,Y) : h(xi)∈clYAifor every i∈ {1, ..., k}};
that is, T= [x0, ..., xk;clYA0, ..., clYAk]∩Cp(X,Y).So, Tis closed in YXand it is dense in i≤kπ−1
xi[clYAi].
Then, T=i≤kπ−1
xi[clYAi]is compact. Hence, each clYAiis compact, x∈i≤nAi⊆i≤nclYAi⊆V, and
either X={x1, ..., xn}or Yis compact and Xis discrete. Moreover, since yand Vwere taken arbitrarily, Yis
locally compact.
Theorem 2.8. Let Ybe a rst countable topological space and Xbe a countable space. Then the following are
equivalent:
1. K(Cp(X,Y)) is a complete lattice,
2. K(Cp(X,Y)) is a lattice,
3. K(Cp(X,Y)) is a b-lattice,
4. Cp(X,Y)is locally compact,
5. Yis locally compact and either Xis nite or Yis compact and Xis discrete.
Proof. Since Xis countable, it is zero-dimensional ([7], Corollary 6.2.8), and so Cp(X,Y)is dense in YX. Then,
χ(Cp(X,Y)) ≤|X|·χ(Y)(see [3], Corollary 4.4), and Cp(X,Y)is rst countable. Thus, by Corollary 2.3 the
statements (1), (2), (3) and (4) are equivalent. The equivalence (4) ⇔(5) is Lemma 2.7.
3k- and strongly k-embedded subspaces
The following result is essential for our purposes. It was already proved in [10] for locally compact spaces.
Theorem 3.1. Let Xand Ybe two homeomorphic spaces. If K(X)is a b-lattice (respectively, lattice, complete
lattice), then K(Y)is a b-lattice (resp., lattice, complete lattice).
Proof. We are only going to prove the theorem for the case when K(X)is a b-lattice. Let h:Y→Xbe a home-
omorphism. Let (bY ,b)be a compactication of Y. There exist a compactication (T,l)of Xand a home-
omorphism H:bY →Tsuch that Hb[Y] = l∘h. Since K(X)is a b-lattice, there are a compact space
K⊆βX \βX[X]and a continuous function f:T→β X/Ksuch that f(l(x)) = βX(x)for all x∈X. Additionally,

The partially pre-ordered set of compactications of Cp(X,Y)|15
there is a homeomorphism φ: (βY,βY)→(βX,βX)such that φβY[Y] = βX∘h. Take M=φ−1[K].Mis a
compact subspace of βY \βY[Y]. We only have to prove that there is a continuous function g:bY →βY/M
such that g(b(y)) = βY(y)for every y∈Y. Dene g(z) = (φ−1∘f∘H)(z). Of course, gis continuous. It re-
mains to prove that gb[Y] = βYY. In fact, if y∈Ythen g(b(y)) = (φ−1∘f)(H(b(y)))) = φ−1(f(l(h(y)))) =
φ−1(βX(h(y))) = βY(y).
Example 3.2. 1. The identity function from the Sorgenfrey line Sto Ris a condensation, K(R)is a b-lattice
but K(S)is not a lattice (see Theorem 1.3).
2. Let E(ωω
1)be the absolute of ωω
1. Because of Theorem 3.3 in [6], K(E(ωω
1)) is a b-lattice but K(ωω
1)is not
a lattice, and ωω
1is a perfect and irreducible image of E(ωω
1).
Recall that a subspace Aof a space Xis a retract of Xif there is a continuous function r:X→Asuch that
r(a) = afor all a∈A. In this case, the function ris called retraction. Observe that if Ais a retract of X, then
Ais C-embedded in X.
Proposition 3.3. Let Sbe a set and for each s∈Slet Xsbe a topological space. Assume that for each s∈S,
Csis a retract of Xs. Then, s∈SCsis a retract in s∈SXs.
Proof. Let rs:Xs→Csbe a retraction for each s∈S. Then the function r:s∈SXs→s∈SCsdened by
r((xs)s∈S) = (rs(xs))s∈Sis a retraction.
Let Xand Ybe two topological spaces and y∈Y. We denote by ctythe constant function that sends all x∈X
to the point y∈Y. The collection of all the constant functions from Xto Yis denoted by Cts(X,Y).
Proposition 3.4. For all spaces Xand Y,Cts(X,Y)is a retract in YX. In particular, Cts(X,Y)is a retract of
Cp(X,Y).
Proof. Let z0∈Xbe xed. The function r:YX→Cts(X,Y)dened by r(f) = ctf(z0)is a retraction.
As we have already agreed, for each locally compact and non compact space Y,αY is the one point compact-
ication of Y. We denote by pYthe point in αY not belonging to Ywhich compacties Y.
Remark 3.5. 1. For every pair of topological spaces Xand Y,Ct s(X,Y)is homeomorphic to Y. Indeed, the
function ϕY:Y→Cts(X,Y)dened by ϕY(y) = c tyis a homeomorphism.
2. If Yis locally compact and non-compact, then
αCts(X,Y) = Cts(X,Y)∪ {ctpY}=Cts(X,αY ).
We can generalize Remark 3.5.(2):
Lemma 3.6. Let Ybe a non-compact space. Let (bY,b)be a compactication of Y. Then, Cts(X,bY) =
cl(bY)XCts(X,b[Y]).
Proof. Since Cts(X,bY)is homeomorphic to bY (Remark 3.5.(1)), Cts(X,bY )is compact. Moreover, Cts(X,b[Y]) ⊆
Cts(X,bY). Thus, cl(bY)XCts(X,b[Y]) ⊆Cts(X,bY ). Let z∈bY \b[Y]. We are going to prove that ctz∈
cl(bY)XCts(X,b[Y]). Let [x1, ..., xn;A1, ..., An]be a canonical open subset of (bY)Xwhich contains ctz. Then
z∈i≤nAi. So, i≤nAiis a non-empty open subset of bY. Thus, we can take a point y0∈i≤nAi∩b[Y]. It
happens then that cty0∈[x1, ..., xn;A1, ..., An]∩Cts(X,b[Y]).
The previous lemma motivates the introduction of the following natural concepts.
Denition 3.7. 1. [5] A subspace Yof a space Xis k-embedded in Xif for each compactication (K,i)of
Y, there exists a compactication (M,j)of Xsuch that (clMj[Y],jY)≤(K,i); that is, there exists a
continuous function h:K→clMYsuch that h(i(y)) = j(y)for every y∈Y.
2. A subspace Yof a space Xis strongly k-embedded in Xif for each compactication (K,i)of Ythere is a
compactication (M,j)of Xsuch that K≡YclMY; that is, there exists a homeomorphism h:K→clMY
such that h(i(y)) = j(y)for every y∈Y. (Observe that the relations of being k-embedded and strongly
k-embedded are transitive.)

16 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
Proposition 3.8. Let X⊆Yand dene φ:K(Y)→K(X)by φ((Z,a)) = (clZa[X],aX).Then, φis a
homomorphism of partially ordered sets with φ[K(Y)] dense in K(X)if and only if Xis k-embedded in Y.
Proof. Assume that Xis k-embedded in Y.We are going to prove that φis a homomorphism. Let (W,a),(Z,b)
be two compactications of Yand suppose that (W,a)≤(Z,b).Then, there exists a continuous function
f:Z→Wsuch that f∘b=a.Observe that f[clZb[X]] ⊆clWf[b[X]] = clWa[X].Therefore, the function
fclZb[X]witnesses that φ((W,a)) ≤φ((Z,b)).
Now we are going to prove that φ[K(X)] is dense in K(Y).Let (W,a)be a compactication of X,since X
is k-embedded in Y,there is a compactication (Z,b)of Ysuch that (clZb[X],bX)≤(W,a).Therefore,
φ((Z,b)) ≤(W,a).
Suppose that φis a dense homomorphism of partially ordered sets, and let (W,a)be a compactication
of X;then there is a compactication (Z,b)of Ysuch that φ((Z,b)) ≤(W,a); this means that (clZb[X],b
X)≤(W,a),therefore Xis k-embedded in Y.
Proposition 3.9. 1. Every closed subset of a locally compact space Xis k-embedded in X.
2. Every closed and C*-embedded subspace of Xis strongly k-embedded in X.
Proof. (1) Let Fbe a closed subset of the locally compact space X. Let (K,j)be a compactication of F. We
claim that clαX F≤(K,j). There are two cases, if clαX F=F, then Fhas to be compact because it remains
closed in αX. In this case K=F. The second possibility happens when clαX F={pX} ∪ F. In this case it is easy
to see that clαX F≡αF and clαX F≤K.
(2) Let Abe a closed and C*-embedded subspace of X. Choose a compactication (K,b)of A. Since Ais C*-
embedded in X, clβ X Ais equivalent to the Stone-Čech compactication of A. Denote by ˜
bthe continuous
extension of bto clβX A,˜
b:clβX A→K. Consider the partition D={{x}:x∈A∪(βX \clβ X A)} ∪ {˜
b−1(y) :
y∈K\b[A]}. Denote by qand Ythe quotient map and the quotient space induced by Don βX, respectively.
Then qidenties two points xand yif and only if x,y∈clβX A\Aand ˜
b(x) = ˜
b(y).
Let us observe that the space Yis compact. We shall verify that Yis a Hausdor space. Choose x,y∈
βX with q(x) =q(y). If x,y∈βX \clβX Achoose disjoint open neighborhoods Uand Vof xand yin βX,
respectively, such that U,V⊆βX \clβX A. Then q(U)and q(V)are disjoint open sets in Ywith q(x)∈q(U)
and q(y)∈q(V). If x∈clβ X Aand y∈βX \clβ X Achoose disjoint open sets Uand Vin βX with clβ X A⊆U
and y∈V. It follows that q(U)and q(V)are disjoint open sets in Ywith q(x)∈q(U)and q(y)∈q(V). The
case x∈βX \clβ X Aand y∈clβX Ais similar. Finally, assume that x,yare both in clβ X A. Let us observe
that q(x) =q(y)implies ˜
b(x) =˜
b(y). Choose open sets U1and V1in Ksuch that ˜
b(x)∈U1,˜
b(y)∈V1
and clU1∩clV1=∅. It follows that cl˜
b−1(U1)∩cl˜
b−1(V1) = ∅. Choose open sets U2and V2in βX with
U2∩clβX A=˜
b−1(U1)and V2∩clβX A=˜
b−1(V1). Finally, let U=U2\(clU2∩clV2)and V=V2\(clU2∩clV2).
Then U∩clβX A=˜
b−1(U1)and V∩clβX A=˜
b−1(V1)and U∩V=∅. Therefore, q(U)and q(V)are disjoint open
neighborhoods of q(x)and q(y)in Y, respectively.
Notice that Aclosed in Ximplies X⊆A∪(βX \clβX A). Then {{x}:x∈X} ⊆ Dand so qXis an
embedding. It follows that (Y,qX)is a compactication of X. We claim that (clYq(A),qA)is equivalent
to (A,b). The map qclβX A:clβ X A→clYq(A)is a quotient map and the function ˜
b:clβX A→Kis constant
on the bers of qclβ X A, then there exists a continuous map d:clYq(A)→Ksuch that ˜
b=d∘qclβX A. Let
us observe that dis one-to-one and hence a homeomorpism. In addition, d(q(x)) = ˜
b(x) = b(x)for any x∈A.
Hence (clY(q(A)),qA)is equivalent to (A,b). So we have proved that Ais strongly k-embedded in X.
Observe that the statement in (1) of the previous proposition cannot be strengthened. An example: Let Abe a
mad family such that βΨ (A) = αΨ(A)where Ψ(A)is the Mrówka-Isbell space dened by A. Then Ais closed
in Ψ(A)but it is not a strongly k-embedded subset of Ψ(A).
Corollary 3.10. Every retract of a space Xis C- and strongly k-embedded in X.
Corollary 3.11. If Xis normal and Ais closed in Xthen Ais strongly k-embedded in X.
Proposition 3.12. Suppose that X⊆Y⊆Z. If Xis k-embedded in Z, then Xis k-embedded in Y.

The partially pre-ordered set of compactications of Cp(X,Y)|17
Proof. Let (K,a)be a compactication of X.There is a compactication (Z′,c)of Zsuch that (clZ′c[X],c
X)≤(K,a).Observe that (clZ′c[Y],cY)is a compactication of Y.Let Y′= clZ′c[Y].Hence clY′c[X] =
clZ′c[X]∩Y′= clZ′c[X].Therefore (clY′c[X],cX)≤(K,a).
In [5] the following result was proved.
Theorem 3.13. If Yis k- and C*-embedded in Xand K(X)is a b-lattice, then K(Y)is a b-lattice.
The converse of Theorem 3.13 is not true: K(Rω)is not a lattice but [0,1]ωis a compact retract of Rω.
We can also obtain:
Theorem 3.14. If Yis homeomorphic to a k-embedded subspace of Xand K(X)is a lattice, then K(Y)is a
lattice.
Proof. Let (M,b)and (N,c)be two compactications of Y. By hypothesis, there is an embedding i:Y→X,
such that i[Y]is k-embedded in X.Hence there are compactications (K,a),(K′,a′)of X,and continuous
functions f:M→clKa[i[Y]],h:N→clK′a′[i[Y]] such that f∘b=a∘iand h∘c=a′∘i.
Since K(X)is a lattice, there exists a compactication (Z,z)of Xand continuous functions h:K→Z,
h′:K′→Zsuch that h∘a=zand h′∘a′=z.
Now observe that h∘f:M→clZz[i[Y]] and h∘g:N→clZz[i[Y]] are continuous functions such that
h∘f∘b=z∘iand h∘g∘c=z∘i.Therefore (clZz[i[Y]],z∘i)≤(M,b)and (clZz[i[Y]],z∘i)≤(N,c).
We conclude that K(Y)is a lattice.
We have already seen that Cts(X,Y)is a retract of Cp(X,Y), so Cts(X,Y)is a strongly k-embedded sub-
space of Cp(X,Y). Hence, we obtain a generalization of Propositions 2.4 and 2.6 above.
Corollary 3.15. Let Ybe a rst countable space which is not locally compact. Then, for every topological
space X,K(Cp(X,Y)) is not a lattice.
Proof. If K(Cp(X,Y)) were a lattice, then K(Y)is a lattice (Theorem 3.1). But Yis rst countable, so by Corol-
lary 2.3, Ymust be locally compact.
Because of Theorem 3.14, we obtain:
Theorem 3.16. Let Xbe a topological space and Ybe a non-compact rst countable space. Then K(Cp(X,Yω))
(and K(Cp(X×ω,Y))) is not a lattice.
Proof. Assume that K(Cp(X,Yω)) is a lattice. Then, it happens that K(Cts(X,Yω)) is a lattice (Theorem 3.14
and Corollary 3.10). By Remark 3.5.(1) and Theorem 3.14, K(Yω) is a lattice. But this last assertion contradicts
Theorem 2.8.
The partially ordered set K(Cp(X×ω,Y)) is not a lattice either because
Cp(X×ω,Y)∼
=Cp(X,Yω).
What can we say when Xhas a dense k-embedded subspace? The following results give us some answers.
Proposition 3.17. Let X⊆Y.If Xis locally compact and dense in Y,then Xis k-embedded in Yif and only if
either Y=Xor Y=αX.
Proof. Assume that Xis k-embedded in Y.There is a compactication bY ∈K(Y),such that clbY X≤Xα X.
Since Xis dense, clbY X=bY and since bY is a compactication of X, then αX ≤XbY .Therefore bY ≡XαX.
Now suppose that Y=αX .Let bX be a compactication of X,then αX ≤b X.
Corollary 3.18. Let X⊆Y.If Xis locally compact and dense in Y,and if Yis compact, then Xis k-embedded
in Yif and only if Y=αX.
Proposition 3.19. Let X⊆Y.If Xis dense in Yand |Y\X|≥2,then Xis not k-embedded in Y.

18 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
Proof. Let x,ybe two distinct points in Y\Xand K={x,y}. Let (aY ,a)be a compactication of Y.Then
(aY /a[K],π∘aX)is a compactication of Xwhere π:aY →aY /a[K]is the natural projection. If Xis k-
embedded in Y,there is a compactication (cY ,c)of Yand there is a continuous function f:aY /a[K]→c Y
such that f∘π∘aX=cX.Observe that f∘π∘a=c,because Xis dense in Y.Since K⊆Y,c(x) =c(y).
But we also have π(a(x)) = π(a(y)),therefore f(π(a(x))) = f(π(a(y))),hence c(x) = c(y),a contradiction.
The converse of the previous result is not true. Indeed, let p∈βω \ω.Observe that ωis dense and C*-
embedded in X=ω∪ {p},but ωis not k−embedded in Xbecause ωis locally compact and Xis not the one
point compactication of ω.
Corollary 3.20. Let A⊆X.If Ais k-embedded in X,then |clXA\A|≤1.
Proof. Since Ais k-embedded in X,Ais k-embedded in clXA(Proposition 3.12), then |clXA\A|≤1.
Proposition 3.21. Let X⊆Y.If Xis dense and k-embedded in Yand if K(X)is a lattice, then K(Y)is a lattice.
Proof. Let (aY ,a),(bY,b)be compactications of Y.Then (aY,aX),(bY,bX)are compactications of
X.Since K(X)is a lattice, there exists a compactication (cX,c)of X, and continuous functions f:aY →cX
and g:bY →cX,such that
f∘aX=cand g∘bX=c.
Since Xis k-embedded in Y,there is a compactication (dY ,d)of Yand there is a continuous function h:
cX →dY = cldY Xsuch that h∘c=dX.Therefore h∘f∘aX=dXand h∘g∘bX=dX.
Now take y∈Y.There is a net (xλ)λ∈Λin Xsuch that limλ∈Λ(xλ) = y.
Hence
h∘f∘a(y) = lim
λ∈Λh∘f∘a(xλ) = lim
λ∈Λd(xλ) = d(lim
λ∈Λ(xλ)) = d(y).
Therefore (dY ,d)≤(aY,a).Analogously (dY ,d)≤(bY,b).Then K(Y)is a lattice.
4˜
k- and k*-embedded subspaces
Theorem 3.14 suggests that we introduce the following denitions.
Denition 4.1. 1. A space Yis ˜
k-embedded in a space Xif there is an embedding j:Y→Xsuch that j[Y]
is k-embedded in X.
2. A space Yis k*-embedded in a space Xif there is an embedding j:Y→Xsuch that j[Y]is C*- and
k-embedded in X.
Example 4.2. There are spaces Xcontaining a subspace Ysuch that Yis ˜
k-embedded in Xbut Yis not k-
embedded in X.
Proof. The interval (0,1) is not k-embedded in R(Propositions 3.9.(1) and 3.17) but it is ˜
k-embedded in R.
Observe, however, that (0,1] is k-embbedded in R.
Remark 4.3. 1. The relation of being ˜
k-embedded is not reexive.
2. The relation of being ˜
k-embedded is transitive.
Proof. Indeed, let X= [0,1) and Y= [0,1].Then Xis ˜
k-embedded in Yand Yis ˜
k-embedded in X. But Xis
not homeomorphic to Y.
Also, the statement (2) is easy to prove.
As a consequence of Lemma 4.12 in [5] we obtain:
Theorem 4.4. Let D,Xand Ybe spaces and y0be an element of Y.If X×{y0} ⊆ D⊆X×Y, and Dis dense
in X×Y, then X×{y0}is C- and k-embedded in D.In particular, X×{y0}is C- and k-embedded in X×Y.
Corollary 4.5. Let Xand Ybe topological spaces. Then Xis k*-embedded in X×Y.

The partially pre-ordered set of compactications of Cp(X,Y)|19
Theorem 4.6. If Xis ˜
k-embedded in Yand Zis homeomorphic to X, then Zis ˜
k-embedded in Y.
Proof. Let i:X→Ybe an embedding witnessing that Xis ˜
k-embedded in Y(that is, i[Y]is k-embedded in
X). Take an homeomorphism h:Z→X. Then i∘h:Z→Yis an embedding and (i∘h)[Z] = i[h[Z]] = i[Y]. So,
(i∘h)[Z]is k-embedded in X. This means that Zis ˜
k-embedded in X.
Corollary 4.7. If f:X→Yis a homeomorphism and A⊆X, then Ais C- and k-embedded in Xif and only if
f[A]is C- and k-embedded in Y.
Related to Theorem 3.13 we have:
Theorem 4.8. If Yis k*-embedded in Xand K(X)is a b-lattice, then K(Y)is a b-lattice.
Proof. There is an embedding j:Y→Xsuch that j[Y]is k- and C*-embedded in X.By Theorem 3.13, K(j[Y])
is a b-lattice, therefore K(Y)is a b-lattice.
The following is a consequence of Theorem 3.14 and Corollary 4.5.
Corollary 4.9. Let Ybe a rst countable space. Let Xbe the free topological sum of two spaces Aand B.
Assume that |A|=ℵ0, and either Ais not discrete or Yis not compact. Then, K(Cp(X,Y)) is not a lattice.
Proof. We have Cp(X,Y)∼
=Cp(A,Y)×Cp(B,Y). Then Cp(A,Y)is ˜
k-embedded in Cp(X,Y), and so if
K(Cp(X,Y)) is a lattice, then K(Cp(A,Y)) is a lattice. But this is not possible because of Theorem 2.8.
Example 4.10. 1. For every rst countable space Yand every ordinal number α>ω,K(Cp([0,α),Y)) is not
a lattice (compare with Proposition 6.8).
2. For every rst countable space Yand every almost disjoint family A,K(Cp(Ψ(A),Y)) is not a lattice.
Proof.1. Indeed, we can apply Corollary 4.9 because
[0,α) = [0,ω]⊕[ω+ 1,α).
2. Let A0be an element of A. It happens that A′=A\(A0∪ {A0})is an independent family in ω\A0. Moreover,
Ψ(A) = (A0∪ {A0})⊕Ψ(A′). Now, we apply Corollary 4.9 and we nish the proof.
Corollary 4.11. Let Ybe a non-compact rst countable space. If Xcontains an innite discrete clopen subset,
then K(Cp(X,Y)) is not a lattice.
5Spaces which are almost the free topological sum of a family of
spaces
Let Xbe the free topological sum of the family {Xs:s∈S}. Let Ybe a topological space. A function f∈YX
such that fis constant in each Xswill be called ladder in YX. The set of ladders in YXwill be denoted by
Lad(X,Y).
Claim (2) in the following lemma is a consequence of Proposition 3.3 and Corollary 3.10.
Lemma 5.1. 1. Let Xbe the free topological sum of the family {Xs:s∈S}. Let Ybe a topological space.
Then La d(X,Y)⊆Cp(X,Y)and the function ϕ:Lad(X,Y)→YSdened by ϕ((ctsrs)s∈S) = (rs)s∈Sis a
homeomorphism.
2. Let Ybe a topological space. Let Xbe the free topological sum of the family {Xs:s∈S}. Then Lad(X,Y)
is C- and k-embedded in Cp(X,Y).
Because of Theorem 3.14 and Corollary 2.3 we have:
Corollary 5.2. Let Ybe a non-compact rst countable space. Let X={Xs:s∈S}with |S|≥ℵ0. Then
K(Cp(X,Y)) is not a lattice.

20 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
Corollary 5.3. Let Ybe a non-compact rst countable space and Xa non pseudocompact zero-dimensional
space. Then, K(Cp(X,Y)) is not a lattice. (Compare to Theorem 6.9 below.)
Example 5.4. For every cardinal number κ,Sκand Nκare examples of zero-dimensional non-pseudocompact
spaces. Then, if Yis a rst countable non-compact space, K(Cp(Sκ,Y)) and K(Cp(Nκ,Y)) are not lattices.
Let κbe an innite cardinal number. A lter Fon κis uniform if |F|=κfor every F∈F. Let Fbe a uniform free
lter on κ. We denote by AF
κthe topological space κ∪ {p}where p∈κ,κis the discrete space of cardinality
κ, and the neighborhoods of pare the subsets Vof AF
κcontaining pand such that V∩κ∈F.
Proposition 5.5. Let Ybe a topological space. Let X=X0∪ {p}where X0={Xs:s∈S},p∈X0and
|S|≥ω. Suppose that there is a uniform free lter Fon Ssuch that the family {{p} ∪ {Xs:s∈F}:F∈F}
is a local base at the point p. Then Cp(X,Y)contains a C- and strongly k-embedded copy of Cp(AF
κ,Y), where
κ=|S|.
Proof. Pick a point ys∈Xsfor each s∈Sand let Z={ys:s∈S}∪{p}. Consider the function r:X→Z
given by r(x) = ysif x∈Xsand s∈Sand r(p) = p. Notice that ris a retraction. Let us observe that Z
is homeomorphic to AF
κ, where κ=|S|. It follows that Cp(Z,Y)is homeomorphic to Cp(AF
κ,Y). Therefore,
Cp(X,Y)contains a C- and strongly k-embedded copy of Cp(AF
κ,Y)(Corollary 3.10).
Example 5.6. Let Ybe a non-compact rst countable space, let κbe an innite cardinal and let Fbe a uniform
free lter on κ. Then, K(Cp(AF
κ,Y)) is not a lattice.
Proof. If Yis not locally compact, we use Corollary 3.15. Assume that Yis locally compact. For κ=ωand F
equal to the Fréchet lter, this proposition has already been proved in Theorem 2.8.
For κ>ωand if Fis not the Fréchet lter on κ, then the conclusion was proved in Corollary 4.11.
Now suppose that κ>ωand Fis the Fréchet lter on κ. In this case, we can consider AF
κas {p}∪⊕i<ωXi
where Xi={as:s∈Si} ⊆ κand i<ωXi=κ,Xiis not empty for every i<ωand Xi∩Xj=∅if i=j. Observe
that for every neighborhood Vof p, there is n0<ωsuch that Xi⊆Vfor all i≥n0. Because of Proposition 5.5,
Cp(AF
κ,Y)contains a C- and k-embedded copy of Cp(AG
ω,Y)where Gis the Fréchet lter on ω. This means
that K(Cp(Aκ,Y)) is not a lattice.
As a consequence of Proposition 5.5 and Example 5.6, we obtain:
Corollary 5.7. Let Ybe a non-compact rst countable space. Let X=X0∪{p}where X0={Xs:s∈S}and
p∈X0with |S|≥ω. Let Fbe a uniform free lter on |S|. Suppose that the family {{p} ∪{Xs:s∈F}:F∈F}
is a local base at the point p. Then K(Cp(X,Y)) is not a lattice.
Let X,Yand Zbe three spaces. If r:X→Zis a continuous function, we denote by r*the continuous function
from Cp(Z,Y)to Cp(X,Y)dened by r*(g) = r∘g.
Theorem 5.8. Let Xand Ybe spaces. Suppose that there exist a set Z⊆Xand a retraction r:X→Z, then
Cp(X,Y)contains a C- and strongly k-embedded copy of Cp(Z,Y).
Proof. Let A=r*[Cp(Z,Y)]. Since r*:Cp(Z,Y)→Cp(X,Y)is an embedding, the map r*:Cp(Z,Y)→Ais
a homeomorphism. We claim that r*∘πZ:Cp(X,Y)→Ais a retraction. Since r:X→Zis a retraction, Z
is C-embedded in X. Then πZ:Cp(X,Y)→Cp(Z,Y)is surjective. It follows that r*∘πZis continuous and
onto. Let g∈A=r*∘πZ[Cp(X,Y)], then gis equal to (fZ)∘rfor some f∈Cp(X,Y). Let us observe that
fZ= ((fZ)∘r)Z. Hence r*∘πZ(g)=(gZ)∘r= (((fZ)∘r)Z)∘r= (fZ)∘r=g. This shows that
r*∘πZis a continuous retraction onto A. It follows from Corollary 3.10 that Ais C- and strongly k-embedded
in Cp(X,Y).
Proposition 5.9. Let X=Z×W. Then, for every space Y,Cp(X,Y)contains a C- and k-embedded copy of
Cp(Z,Y).

The partially pre-ordered set of compactications of Cp(X,Y)|21
Proof. Pick w∈Wand let Z*=Z×{w}. The function r:X→Z*given by r((z,w)) = (z,w)is a retraction. By
Theorem 5.8, Cp(X,Y)contains a C- and k-embedded copy of Cp(Z*,Y). The result follows from the fact that
Zand Z*are homeomorphic.
Example 5.10. We can nd two spaces Xand Ywhich contain k-embedded subsets Aand B, respectively,
and however A×Bis not k-embedded in X×Y. Indeed, let Abe a mad family on ω, then ω×Ψ(A)is not
k-embedded in ω×αΨ(A). The reason for this is that every dense and k-embedded weakly pseudocompact
subspace Zin a space Wreects the weak pseudocompactness to W(Theorem 6.4 in [5]), ω×Ψ(A)is weakly
pseudocompact but ω×αΨ (A)is Lindelöf and locally compact, so it is not weakly pseudocompact. Then,
ω×Ψ(A)is not k-embedded in ω×α Ψ (A).
The following result is easy to prove.
Proposition 5.11. Assume that r:Y→Zis a retraction. Then, r*:Cp(X,Y)→Cp(X,Z)is a retraction, where
r*(f) = r∘f.
Corollary 5.12. If K(Cp(X)) is a lattice (resp., b-lattice), then K(Cp(X,[0,1])) is a lattice (resp., b-lattice). And
if K(Cp(X,Z)) is a lattice (resp., b-lattice), then K(Cp(X,{0,1})) is a lattice (resp., b-lattice).
Proof. The space [0,1] (resp., {0,1}) is a retract of R(resp., Z).
The converse of Corollary 5.12 is not true. Indeed, [0,1]ω(resp., {0,1}ω) is compact but K(Rω)(resp., K(Zω))
is not a lattice.
6The upper semilattice K(Cp(X))
Given a space Xand a closed subset Fof Xwe denote by ZF(X)the subspace {f∈Cp(X) : f(F)⊂ {0}} of
Cp(X). If F={x} ⊂ Xwe denote ZF(X)simply as Zx(X).
Proposition 6.1. Let Fbe a uniform free lter on κ. Then K(Zp(AF
κ)) is not a lattice (see the notation before
Proposition 5.5).
Proof. Indeed, if κ=ω,K(Zp(AF
ω)) is not a lattice since Zp(AF
ω)is rst countable and non-locally compact.
If κ>ωand Fis not the Fréchet lter, then Zp(AF
κ)∼
=Rω×Zp(AFκ\ω
κ\ω). By Theorem 4.4, Zp(AF
κ)contains a
k-embedded copy of Rω. So, in this case, K(Zp(AF
κ)) is not a lattice either.
Now assume that κ>ωand Fis the Fréchet lter. Consider a partition {Kn:n<ω}of κwhere each
Knis not empty. Choose a point xn∈Knfor each n<ω. Now, we can follow a similar argumentation to that
given in the proof of Proposition 5.5, and conclude that Zp(AF
κ)contains a C- and k-embedded copy of Zp(AG
ω)
where Gis the Fréchet lter on ω. Therefore, also in this last possible case K(Zp(AF
κ)) is not a lattice.
Theorem 6.2. Let Fbe a uniform free lter on κ. If Xcontains a subspace X0={s} ∪ ξ<ωUξsuch that each
Uξis open in X, the family {clXUξ:ξ<κ}is a pairwise disjoint family, {ξ<κ:Uξ⊆V} ∈ Ffor every
neighborhood Vof sin X, and {s}= clX(ξ<κUξ)\ξ<κclXUξ, then
1. the space Cp(X)contains a retract homeomorphic to Zp(AF
κ),
2. the space Cp(X)is homeomorphic to Zp(AF
κ)×Yfor some space Y.
Proof. (1) Pick sξ∈Uξ. Then, S={s} ∪ {sξ:ξ∈κ} ⊆ Xis homeomorphic to AF
κ. It follows that Zs(S)
is homeomorphic to Zp(AF
κ). For each ξ∈κ, we can pick a continuous function fξ:X→[0,1] such that
fξ(sξ) = 1 and fξ(X\Uξ) = {0}. Dene a function e:Z→Cp(X)by
e(g)(x) = g(sξ)fξ(x)if x∈clXUξfor some ξ∈κ;
0otherwise.
The following claim shows that eis well dened.

22 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
Claim 1. The function e(g)is continuous for any g∈Zs(S).
Pick g∈Zsω(S). Choose x∈Xand an open set Vin Rcontaining e(g)(x). We shall prove that there exists
an open set Uin Xwith x∈Uand e(g)(U)⊂V. Consider three cases.
Case 1. If x∈ξ<κclXUξ\{s}, let Uηbe such that x∈clXUη. We have that e(g)clXUη= (n∈ωg(sξ)fξ)
clXUη=g(sη)fηclXUη. Let Abe an open set of Rcontaining e(g)(x) = g(sη)fη(x). Since g(sη)fη:X→R
is a continuous function in x, there is an open subset B′of Xsuch that x∈B′and g(sη)fη[B′]⊆A. Let
B=B′∩X\({s} ∪ ξ=ηclXUξ). Since {s} ∪ ξ=ηclXUξis closed, Bis open, contains xand is contained in
B′. Observe that e(g)B=g(sη)fηB; so, e(g)[B] = g(sη)fη[B]⊆g(sη)fη[B′]⊆A. We conclude that e(g)is
continuous in x.
Case 2. If x=s. Let Abe an open set in Rwhich contains [e(g)](s)=0. Choose ϵ>0with (−ϵ,ϵ)⊆A.
Since g(s) = 0 and gis continuous in s, we can nd F∈Fsuch that g(sξ)∈(−ϵ,ϵ)if and only if ξ∈F.
Let Vs=X\clX(ξ∈FUξ). Clearly Vsis open and contains s. We will prove that [e(g)](Vs)⊆A. Pick y∈Vs.
Either y∈{cl(Uξ) : ξ∈κ}or there exists a unique η∈Ffor which y∈cl(Uη). Hence either e(g)(y)=0or
[e(g)](y) = ξ∈κg(sξ)fξ(y) = g(sη)fη(y). In both cases we conclude that [e(g)](y)∈(−ϵ,ϵ). Hence [e(g)](Vx)⊆
(−ϵ,ϵ)⊆A. This shows that e(g)is continuous in s.
Case 3. Now, let xbe an element in X\clXξ<κUξ. Take Ux=X\clXξ<ωUξ. It happens that Uxis open, x∈Ux
and e(g)(Ux) = {0}={e(g)(x)}. Thus, for every open subset Aof Rwhich contains 0,e(g)[Ux] = {0} ⊆ A.
Claim 2. The map e:Zs(S)→Cp(X)is an embedding.
Let P=e(Z). It is easy to verify that e−1=πSPand hence e−1is continuous. So, in order to prove that
eis an embedding we shall only prove that eis continuous. Pick g∈Zs(S)and an open set Vin Cp(X)which
contains e(g). We can suppose that Vis a subbasic open set, that is V= [x,B]∩Cp(X)where x∈Xand Bis
an open subset of R. If x∈X\{f−1
ξ((0,1]) : ξ∈κ}choose W=Zs(S). It is clear, in this case, that g∈W
and e(W)⊆V. Now suppose, otherwise, that x∈{f−1
ξ((0,1]) : ξ∈κ}. Then there exists a unique ordinal
number ηwith x∈f−1
η((0,1]). It follows from e(g)∈Vthat g(sη)fη(x) = n∈ωg(sξ)fξ(x)=[e(g)](x)∈B.
Since the function m:R→Rgiven by m(r) = rf k(x)is continuous, we can nd an open subset B′of Rsuch
that g(sη)∈B′and m(B′)⊆B. Let W= [sη,B′]∩Cp(S)∩Zs(S). Clearly g∈W. Moreover, for h∈W, it follows
from h(sη)∈B′that [e(h)(x)] = ξ∈κh(sξ)fξ(x) = h(sη)fη(x)∈m(B′)⊆B. Hence in this case we also have
e(W)⊆V. Therefore, eis continuous and so we have proved Claim 2.
Let us observe that the map e∘πS:Cp(X)→Pis a continuous retraction. Since the space Pis homeo-
morphic to Zs(S), we have proved (1).
(2) The space Cp(X)is homeomorphic to the product Zs(X)×R. Indeed, the map φ:Cp(X)→Zs(X)×Rgiven
by φ(g)=(g−g(s),g(s)) is clearly a homeomorphism. Notice that Zs(X)is a topological subgroup of Cp(X)
and e[Z] = P⊆Zs(X). Consider the function l=e∘πSY:Y→P. Observe that lis a continuous and
linear retraction onto P. Let q:Y→Ybe the function given by q(f) = f−l(f)for any f∈Zs(X). Then qis a
continuous function. Let Q=q(Zs(X)). Notice that l(q(f)) = l(f−l(f)) = l(f)−l(l(f)) = l(f)−l(f)=0for any
f∈Zs(X). We claim that Zs(X)is homeomorphic to P×Q. Consider the function ϕ=p∆q :Y→P×Q. We
shall prove that ϕis a homeomorphism. Assume that (l(f),q(g)) in P×Q. Let h=l(f) + q(g)∈Zs(X). Then
l(h) = l(l(f) + q(g)) = l(l(f)) −l(q(g)) = l(f)and q(h) = h−l(h) = (l(f) + q(g)) −l(f) = q(g). So ϕ(h) = (l(f),q(g)).
It follows that ϕis an onto function. It is clear that ϕis continuous. Finally let us observe that ϕ−1=aP×Q,
where a:Cp(X)×Cp(X)→Cp(X)is the sum function. Indeed, a(ϕ(f)) = a((l(f),q(f))) = l(f) + q(f) = l(f)+(f−
l(f)) = f. Thus ϕ−1is continuous and hence ϕis a homeomorphism. Then Zs(X)is homeomorphic to P×Q.
Since Pis homeomorphic to Zp(AF
κ), the space Zs(X)is homeomorphic to Zp(AF
κ)×Q. It follows that Cp(X)is
homeomorphic to Zp(AF
κ)×Q×R, and so we have proved (2).
Corollary 6.3. K(Cp(X)) is not a lattice when Xcontains a subspace X0={s} ∪ ξ<κUξsuch that the family
{Uξ:ξ<κ}is a pairwise disjoint family of open subsets of X, and there is a uniform free lter Fon κsuch
that for every neighborhood Vof sin X,{ξ<κ:Uξ⊆V} ∈ F, and {s}= (clXξ<κUξ)\ξ<κclXUξ.

The partially pre-ordered set of compactications of Cp(X,Y)|23
Proof. If K(Cp(X)) is a lattice, so is K(Zp(AF
κ)). This contradicts Proposition 6.1.
Corollary 6.4. If Xcontains a non-isolated point of countable character, then K(Cp(X)) is not a lattice.
Proof. Let sbe a non-isolated point of Xwith countable character. Then there exists a countable family {Bn:
n∈ω}of open sets in Xwhich is a local base at the point s. We can assume that clBn+1 is a proper subset of
Bnfor every n∈ω. For each n∈ωpick a non-empty open set Unwith Un⊆clUn⊆Bn\clBn+1. Then X0=
{s} ∪ i<ωUi⊆Xand the Fréchet lter Fon ωsatisfy the conditions requested in Corollary 6.3. Therefore,
K(Cp(X)) is not a lattice.
Proposition 6.5. If r:X→Xis a retraction and r(X)is innite and rst countable at some non-isolated point,
then K(Cp(X)) is not a lattice.
Proof. Assume that r[X]is rst countable at some point. Then, by Corollary 6.4, K(πr(X)(Cp(X))) is not a lattice.
Choose ˜
r=r*∘πr(X):Cp(X)→Cp(X). As in the proof of Theorem 5.8 we can show that ˜
ris a retraction.
Moreover, since r*is an embedding, ˜
r(Cp(X)) is homeomorphic to πr(X)(Cp(X)). Hence K(˜
r(Cp(X))) is not a
lattice. By Corollary 3.10 the space ˜
r(Cp(X)) is k-embedded in Cp(X). Therefore, K(Cp(X)) is not a lattice.
Corollary 6.6. K(Cp(R)) is not a lattice.
Corollary 6.7. K(Cp(Cp(X))) is never a lattice.
Proof. Let Xbe an arbitrary space. The space Cp(X)is homeomorphic to the product Y×R, where Y={f∈
Cp(X) : f(x0) = 0}is a subspace of Cp(X)and the point x0∈Xis chosen arbitrarily. Then Cp(Cp(X)) is
homeomorphic to Cp(Y×R). By Proposition 5.9 the space Cp(Y×R)contains a C- and k-embedded copy of
Cp(R). So we can apply Corollary 4.7 to see that Cp(Cp(X)) also contains a C- and k-embedded copy of Cp(R).
Finally apply Theorem 3.14 and Corollary 6.6 to see that K(Cp(Cp(X))) is not a lattice.
If Xis a generalized ordered topological space, there exist a complete linearly ordered topological space

Xcontaining Xdensely and such that the topology inherited by Xfrom 
Xcoincides with the topology of X. For
a cardinal number κ, we say that an increasing sequence (xξ)ξ<κin 
Xis precise if xξ<xηwhen ξ<η<κ, and
for each ordinal limit η<κ,xη=sup{xξ:ξ<η}. In a similar way we dene a precise decreasing sequence
(xξ)ξ<κin 
X. For each point x∈Xwe dene lt(x) = min{κ:there is a precise increasing sequence (xξ)ξ<κin 
X
such that x=sup{xξ:ξ<κ}}; and we dene rt(x) = min{κ:there is a precise decreasing sequence (xξ)ξ<κ
in 
Xsuch that x=inf{xξ:ξ<κ}.
Proposition 6.8. If Xis an innite generalized ordered topological space, then K(Cp(X)) is not a lattice.
Proof. If Xis discrete, the conclusion follows from Theorem 1.3. Assume that Xis not discrete. There is a
linearly ordered topological space 
Xwhich is complete and such that Xis a dense subspace of 
X. Dene
A={x∈X:lt(x)is innite}and B={x∈X:rt(x)is innite}. Since Xis not discrete at least one of the
sets Aand Bis non-empty. Say Ais not empty. Let z∈Asuch that lt(z) = inf{lt(x) : x∈A}. We write simply
κinstead of lt(z). Observe that κis a regular cardinal. For each limit ordinal η<κconsider the following
sequence of non-empty open subsets of X(recall that Xis a dense subspace of 
X):
Aη= (xη,xη+1)∩X, ..., Aη+n= (xη+2n,xη+2n+1 )∩X, ....
Claim: The family {clXAξ:ξ<κ}is a pairwise disjoint family, {ξ<κ:Aξ⊆V} ∈ Ffor every neighborhood
Vof zin Xwhere Fis the free lter on κgenerated by all the nal segments of κ, and {z}= clX(ξ<κAξ)\
ξ<κclXAξ,
If ξ0=η0+n0<ξ1=η1+n1where η0and η1are limit ordinals, then clXAξ0⊆[η0+ 2n0,η0+ 2n0+ 1]
and clXAξ1⊆[η1+ 2n1,η1+ 2n1+ 1] and
[η0+ 2n0,η0+ 2n0+ 1] ∩[η1+ 2n1,η1+ 2n1+ 1] = ∅.

24 |A. Dorantes-Aldama, R. Rojas-Hernández, and Á. Tamariz-Mascarúa
So the collection {clXAξ:ξ<κ}is a pairwise disjoint family. On the other hand, let a,b∈
Xsuch that
a<z<b. By hypothesis, there is ξ<κsuch that a<x𝛾<zfor every 𝛾∈κsuch that ξ<𝛾. Hence
A𝛾⊆(a,b)for every 𝛾>ξ. Then, for every neighborhood Vof zin Xwe have: {ξ<κ:Aξ⊆V} ∈ F. Finally,
clX(ξ<κAξ)\ξ<κclXAξ⊆ {z} ∪ {xη:η<κand ηis a limit ordinal}. If xηbelongs to X, then lt(xη)<lt(z)
and this contradicts the election of z. Therefore, {z}= clX(ξ<κAξ)\ξ<κclXAξ.
Because of Claim and Corollary 6.3, we conclude that K(Cp(X)) is not a lattice.
Theorem 6.9. Let Xbe a non-pseudocompact space. Then K(Cp(X)) is not a lattice.
Proof. Indeed, we are going to prove that in this case Cp(X)contains a k*-embedded copy of Rω. We use a
similar proof to that of Theorem 6.2.
Since Xis not pseudocompact, there is a collection {Un:n∈ω}of open subsets of Xsuch that clXUn∩
clXUm=∅for every n,m<ωwith m=n, and n<ωclXUn= clXn<ωUn. Pick sn∈Un. Then, S={sn:n∈
ω} ⊆ Xis homeomorphic to ω. Let Z=Cp(S). For each n∈ω, we can pick a continuous function fn:X→
[0,1] such that fn(sn) = 1 and fn(X\Un) = {0}. Dene a function e:Z→Cp(X)by e(g) = n∈ωg(sn)fn.
Following similar arguments as those given in the proof of Theorem 6.2, we obtain that e(g)is continuous for
any g∈Z, and eis an embedding.
If X=n<ωclXUn, then we have the conclusion of our theorem as a consequence of Corollary 5.2. Now,
assume that X=n<ωclXUnand let s∈X\n<ωclXUn. Let P=e[Z]. We continue the proof as expected:
the space Cp(X)is homeomorphic to the product Y×R, where Y={f∈Cp(X) : f(s)=0}is a subspace of
Cp(X). Notice that Yis a topological subgroup of Cp(X)and P⊆Y. The function p=e∘πSY:Y→Pis a
retraction onto P. The function q:Y→Ygiven by q(f) = f−p(f)for any f∈Yis continuous. Let Q=q(Y).
Yis homeomorphic to P×Qand hence Cp(X)is homeomorphic to P×Q×R.
Finally, following similar arguments to those given at the end of the proof of Theorem 6.2, we conclude
that Cp(X)contains a C- and k-embedded copy of Rω. Therefore, K(Cp(X)) is not a lattice.
For pseudocompact spaces, we have:
Theorem 6.10. If Xis an innite pseudocompact space and Cp(X)is normal, then K(Cp(X)) is not a lattice.
Proof. Let Sbe a countable innite subset of Xand let Tbe the family of all one-to-one maps from 2to S. For
each t∈Tchoose a continuous map ft:X→Rsuch that ft(t(0)) =ft(t(1)). Let f:X→RTbe the diagonal of
the family of maps {ft:t∈T}. Let us observe that fSis injective. Then Y=f(X)⊂RTis an innite second
countable compact space. It was proved in [1] that every continuous surjective map from a pseudocompact
space to a second countable space is an R-quotient. Then fis an R-quotient, that is f*(Cp(Y)) is a closed
subspace of Cp(X). Since Cp(X)is normal, we can apply Corollary 3.11 to see that f*(Cp(Y)) is strongly k-
embedded in Cp(X). Since Yis an innite second countable compact space, it follows from Corollary 6.4 that
K(Cp(Y)) is not a lattice. Finally we can apply Theorem 3.14 to see that K(Cp(X)) is not a lattice.
Corollary 6.11. If Fis a closed innite subspace of a Σ-product of second countable spaces, then K(Cp(F)) is
not a lattice.
Proof. If Fis not pseudocompact the result follows from Theorem 6.9. If Xis pseudocompact, it is known that
Cp(X)is normal, so the result follows from Theorem 6.10.
In particular:
Corollary 6.12. If Xis an innite Corson compact space, then K(Cp(X)) is not a lattice.
7Open problems
A zero-dimensional space Xis b2-discrete if every countable subspace Nof Xis closed, discrete and every
function f:N→{0,1}can be extended to a continuous function to all X.

The partially pre-ordered set of compactications of Cp(X,Y)|25
Problem 7.1. Are there spaces Xand Yfor which Cp(X,Y)is dense in YXand such that K(Cp(X,Y)) is a
b-lattice? In particular, is there a b2-discrete space Xsuch that K(Cp(X,2)) is a b-lattice?
Problem 7.2. With respect to Example 4.10, is K(Cp([0,ω1),Y)) or K(Cp(Ψ(A),Y)) a lattice, when Ais a mad
family on ωand Y∈ {2,2ω,[0,1]}?
Problem 7.3. Regarding Example 5.4, is K(Cp(Sκ,Y)) or K(Cp(Nκ,Y)) a lattice, when Y∈ {2,2ω,[0,1]}?
Problem 7.4. Is K(Cp(AF
κ,Y)) a lattice for an innite cardinal κand a uniform free lter Fon κ, when Y∈
{2,2ω,[0,1]}? See Example 5.6.
Problem 7.5. Is it true that if Ais k-embedded in Cp(X)and Bis k-embedded in Cp(Y), then A×Bis k-
embedded in Cp(X)×Cp(Y)?
Problem 7.6. State if the following proposition is true: K(Cp(X)) is a lattice i K(Cp(X)) is a b-lattice.
The two last problems must be emphasized on this list because of their importance.
Problem 7.7. Is it true that K(Cp(X)) is not a lattice when Xcontains a convergent sequence?
Problem 7.8. Is K(Cp(X)) not a lattice when Xis not a countably compact space?
Acknowledgement: The authors would like to thank the referee for her/his guiding and stimulating com-
ments. The rst author was supported by CONACyT grant No. 178425/245149. The research of the third author
was supported by PAPIIT IN115312.
References
[1] Arhangelskii A.V., On R-quotient mappings of spaces with a countable base