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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 14, no. 1, 2013

pp. 33-40

Range-preserving AE(0)-spaces

W. W. Comfort and A. W. Hager

Abstract

All spaces here are Tychonoff spaces. The class AE(0) consists of
those spaces which are absolute extensors for compact zero-dimensional
spaces. We define and study here the subclass AE(0)rp, consisting of
those spaces for which extensions of continuous functions can be cho-
sen to have the same range. We prove these results. If each point of
T ∈ AE(0) is a Gδ-point of T , then T ∈ AE(0)

rp. These are equiva-
lent: (a) T ∈ AE(0)rp; (b) every compact subspace of T is metrizable;
(c) every compact subspace of T is dyadic; and (d) every subspace of
T is AE(0). Thus in particular, every metrizable space is an AE(0)rp-
space.

2010 MSC: Primary 54C55. Secondary 06F20, 46E10, 54E18

Keywords: absolute extensor; retraction; zero-dimensional space; range-
preserving function; Dugundji space; dyadic space; countable
chain condition

1. Preliminaries

All spaces here are assumed Tychonoff.
For spaces X and Y , the symbol C(X, Y ) denotes the set of continuous

functions from X into Y .
We write Y ⊆h X to indicates that X contains a homeomorph of Y .
Let X be a homeomorphism-closed class of spaces. Then AE(X) [resp.,

AE(X)rp], the class of absolute extensors [resp., range-preserving absolute ex-
tensors] for X, consists of those spaces T for which, whenever X ∈ X and F
is a closed subset of X, every f ∈ C(F, T ) extends to f ∈ C(X, T ) [resp., and
with f[X] = f[F ]].



34 W. W. Comfort and A. W. Hager

For X a class of spaces, we write

PX := {Πi∈I Xi : i ∈ I ⇒ Xi ∈ X}.

It is clear for arbitrary X, since πi ◦ f ∈ C(F, Ti) for each space T = Πi∈I Ti
and f ∈ C(X, T ), that

(1.1) PAE(X) = AE(X) for every class X.

We note below in Theorem 1.5((a) and (d)) that the relation PAE(X)rp =
AE(X)rp can fail—indeed it fails when X = 0, the class of compact zero-
dimensional spaces. The class AE(0) has been much studied; see [1] for in-
formation and extensive bibliographic citations. In this paper we focus on its
subclass AE(0)rp, which so far as we know is defined and studied for the first
time here.

The class of compact spaces in AE(0) has been intensively studied. Accord-
ing to Haydon [10], it coincides with the class of Dugundji spaces as defined by
Pe lczyński [14], and the subclass 0 ∩ AE(0) of AE(0) coincides with the class
of Stone spaces of projective Boolean algebras ([13]).

Let 2 denote the two-point discrete space.
We begin with a simple basic observation.

Theorem 1.1. 2 ∈ AE(0)rp.

Proof. Let f ∈ C(F, 2) with F closed in X ∈ 0. If f is a constant function
then surely f extends to f ∈ C(X, 2) with f[X] = f[F ], so we assume Fi :=
f−1(i) 6= ∅ for i ∈ 2. For x ∈ F0 there is a clopen neighborhood Ux of x ∈ X
such that Ux ∩ F1 = ∅, and since F0 is compact some finitely many of the sets
Ux (x ∈ F0) cover F0. The union U of those sets covers F0, is clopen in X, and
is disjoint from F1, and then the function f ∈ C(X, 2) defined by

f ≡ 0 on U, f ≡ 1 on X\U
extends f as required. �

From [4](6.2.16) we have for each space X that X is zero-dimensional if and
only if there is a cardinal κ such that X ⊆h 2

κ. It follows then quickly from
(1.1) that

(1.2) AE(0) = AE(P({2}), hence AE(0)rp = AE(P{2})rp.

For a qualitative distinction between the class AE(0) and its subclass AE(0)rp,
one may compare the equivalence (a) ⇔ (b) of the following Theorem with the
fact that every (Tychonoff) space Y embeds as a subspace of a (compact) space
T ∈ AE(0). (To see that, recall that [0, 1] ∈ AE(0) by the classical Tietze-
Urysohn extension theorem, and that Y embeds into some T ∈ P{[0, 1]}; then,
use (1.1) with X = {[0, 1]}.)

Theorem 1.2. For each space T , these conditions are equivalent.
(a) T ∈ AE(0)rp;
(b) S ⊆ T ⇒ S ∈ AE(0)rp; and
(c) S ⊆ T , S compact ⇒ S ∈ AE(0)rp.



Range-preserving AE(0)-spaces 35

Proof. (a) ⇒ (b). Given such S and T and f ∈ C(F, S) with F closed in
X ∈ 0, there is f ∈ C(X, T ) such that f ⊆ f and f[X] = f[F ] ⊆ S.

(b) ⇒ (c). This is obvious.
(c) ⇒ (a). Given f ∈ C(F, T ) with closed F ⊆ X ∈ 0, the space S := f[F ]

is compact. Thus there is f ∈ C(X, T ) such that f ⊆ f and f[F ] = f[F ] = S.
This shows T ∈ AE(0)rp. �

In Theorem 1.5 we make additional simple observations which highlight
differences between the classes AE(0) and AE(0)rp. For that, these definitions
will be useful.

Definition 1.3. Let T be a space.

(a) T is a countable chain condition space (briefly, a c.c.c. space) if every
family of pairwise disjoint open subsets of T is countable.

(b) T is dyadic if for some cardinal κ there is a continuous surjection from
2κ onto T .

Lemma 1.4. (a) Every compact T ∈ AE(0) is dyadic.
(b) Every dyadic space is a c.c.c. space.

Proof. (a) As with every compact space, there exist a cardinal κ, a closed
subspace F of 2κ, and a continuous surjection f : F ։ T ([4](3.2.2)). Since
2κ ∈ 0 and T ∈ AE(0) there is f ∈ C(2κ, T ) such that f ⊇ f. Then f[2κ] = T .

(b) It is well known ([4](2.3.18)) that every product of separable spaces—in
particular, the space 2κ—is a c.c.c. space; and the c.c.c. property is preserved
under continuous surjections. �

Here and later we denote by αD the one-point compactification of the dis-
crete space D of cardinality ℵ1.

Theorem 1.5. Let κ be a cardinal.
(a) 2 ∈ AE(0)rp;
(b) 2κ ∈ AE(0);
(c) if κ > ℵ0 then there is compact T ⊆ 2

κ such that T /∈ AE(0);
(d) if κ > ℵ0 then 2

κ /∈ AE(0)rp.

Proof. (a) was noted in Theorem 1.1, and (b) follows from (1.1) since AE(0)rp ⊆
AE(0).

It is easily seen, as in [4](6.2.16), that αD ⊆h 2
ℵ1 . Clearly the (compact)

space αD is not a c.c.c. space, so αD /∈ AE(0) by Lemma 1.4. That shows
(c), and (d) follows from Theorem 1.2. �

The gist of Theorem 1.5 is that while the class AE(0)rp is “completely
hereditary” (Theorem 1.2), the class AE(0) is not even compact-hereditary;
and AE(0), like every class AE(X), is completely productive (1.1), while the
class AE(0)rp is not even ℵ1-productive. We will see in Corollary 2.5 below
that AE(0)rp is (exactly) countably productive.



36 W. W. Comfort and A. W. Hager

2. Characterizing the Spaces in AE(0)rp

Our principal results about the class AE(0)rp are given in Theorems 2.1 and
2.2 and its corollaries.

Theorem 2.1. If T ∈ AE(0) and each point of T is a Gδ-point, then T ∈
AE(0)rp.

Proof. Given f ∈ C(F, T ) with F closed in X ∈ 0 and T ∈ AE(0), we must
find f ∈ C(X, T ) such that f ⊆ f and f[X] = f[F ]. Since X ∈ 0 there is κ ≥ ω
such that X ⊆h 2

κ, and since 2κ ∈ 0 and T ∈ AE(0) there is f∗ ∈ C(2κ, T )
such that f ⊆ f∗. Then, since 2 is a separable space and points of T are
Gδ-points, the function f

∗ factors through a countable subproduct of 2κ in the
sense that there exist countable C ⊆ κ and g ∈ C(2C, T ) such that f∗ = g ◦ πC
(with πC the usual projection πC : 2

κ
։ 2C). (The theorem just used, due

to A. Gleason, is stated and proved in detail by Isbell [12](p. 132).) Since
2C is compact metrizable, its continuous image g[2C] is compact metrizable
([4](3.1.28)). Then since f[F ] is closed in the separable, zero-dimensional,
metrizable space g[2C ], there is a (continuous) retraction r : g[2C] ։ f[F ] (see
[4](6.2.B) for a proof of this assertion, credited by Engelking to Sierpiński [15]).
Then f := r ◦ f∗|X = r ◦ g ◦ πC|X is as required. In detail:

(1) f∗ is defined on 2κ, so f is well-defined on X;
(2) x ∈ X ⇒ f(x) = (r ◦ g ◦ πC )(x) ∈ r[g[2

C ]] ∈ f[X]; and
(3) x ∈ F ⇒ f(x) ∈ f[F ], so f(x) = (r ◦ g ◦ πC )(x) = r(f(x)) = f(x). �

Theorem 2.2. For each space T , these conditions are equivalent.
(a) T ∈ AE(0)rp;
(b) each compact subspace of T is dyadic;
(c) each compact subspace of T is metrizable.

Proof. (a) ⇒ (b). If compact S ⊆ T , then S ∈ AE(0)rp ⊆ AE(0) by Theo-
rems 1.2 and 1.5, so (b) holds by Lemma 1.4.

(b) ⇒ (c). Suppose that some compact S ⊆ T is nonmetrizable, so that
w(S) = κ > ℵ0. Then, since S is dyadic, some point of S has local weight
(character) κ (by a theorem of Esenin-Vol′pin [5], cited in [4](3.12.12(e))).
Then S contains a copy of the one-point compactification of the discrete space
of cardinality κ (by a theorem of Engelking [3], cited in [4](3.12.12(i))). Then
S contains the (compact, non-c.c.c.) space αD. Since αD is not dyadic (by
Lemma 1.4(b)), the assumption w(S) > ℵ0 is false so S is metrizable.

(c) ⇒ (a). According to Theorem 1.2((a) ⇒ (b)), it suffices to show for
each compact S ⊆ T that S ∈ AE(0)rp. Given such S, from (c) we have
S ⊆h [0, 1]

ω with [0, 1]ω ∈ AE(0) by (1.1) (since surely [0, 1] ∈ AE(0)), so
S ⊆ [0, 1]ω ∈ AE(0)rp by Theorem 2.1. Then S ∈ AE(0)rp, as required. �

It is immediate from Theorem 2.2 that a compact space is closed-hereditarily
dyadic if and only if it is metrizable. That is a result of Efimov [2], reproved
in [3](p. 300).

Corollary 2.3. Every metrizable space is an AE(0)rp-space.



Range-preserving AE(0)-spaces 37

Corollary 2.4. For each space T , these conditions are equivalent.
(a) T ∈ AE(0)rp;
(b) S ⊆ T ⇒ S ∈ AE(0);
(c) S ⊆ T , S closed ⇒ S ∈ AE(0); and
(d) S ⊆ T , S compact ⇒ S ∈ AE(0).

Proof. That (a) ⇒ (b) is clear, since AE(0)rp ⊆ AE(0) and the class AE(0)rp

is hereditary.
That (b) ⇒ (c) and (c) ⇒ (d) are obvious.
If (d) holds then by Lemma 1.4(a) every compact S ⊆ T is dyadic and

Theorem 2.2((b) ⇒ (a)) gives (a). �

Corollary 2.5. Let {Ti : i ∈ I} be a set of nonempty spaces and set T :=
Πi∈I Ti. Then T ∈ AE(0)

rp if and only if
(i) each Ti ∈ AE(0)

rp, and
(ii) |{i ∈ I : |Ti| > 1}| ≤ ℵ0.

Proof. “only if”. Each Ti ⊆h T , so Theorem 1.2((a) ⇒ (b)) shows (i). If (ii)
fails then 2ℵ1 ⊆h T , and then from 2

ℵ1 /∈ AE(0)rp (Theorem 1.5(d)) would
follow the contradiction T /∈ AE(0)rp (from Theorem 1.2).

“if”. We assume without loss of generality that |I| ≤ ℵ0. By Theorem 2.2,
it suffices to show that each compact S ⊆ T is metrizable. Given such S we
have for each i ∈ I that the (compact) space πi[S] is metrizable, so Πi∈I πi[S]
(and hence its subspace S) is metrizable. �

We continue with additional corollaries of the foregoing theorems. In Corol-
lary 2.7 we note that a number of familiar spaces are in the class AE(0)rp, and
in Corollary 2.8 we show that spaces which are “locally in AE(0)rp” are in fact
in AE(0)rp. (That result is in parallel with the theorem from [14] that “locally
Dugundji” implies Dugundji; the converse to that result is given by Hoffmann
[11].)

We first remind the reader of the relevant definitions.

Definition 2.6. Let T = (T, T ) be a space.

(a) A network in T is a family N of subsets of T such that if x ∈ U ∈ T
then there is N ∈ N such that x ∈ N ⊆ U;

(b) T is a σ-space if it has a σ-discrete network;
(c) T is a P -space if every Gδ-subset of T is open.

We refer the reader to [7], especially (§4), for a useful introduction to σ-
spaces. It is noted there, for example, that every Moore space (in particular,
every metrizable space and every countable space), is a σ-space; further, every
(countably) compact subspace of a σ-space is metrizable ([7](p. 447)).

Every compact subspace of a P -space, being finite ([6](4K)), is metrizable.
Using those facts, or otherwise, we have the following corollary to Theo-

rem 2.2((c) ⇒ (a)).

Corollary 2.7. Every σ-space, and every P-space, and every countable space,
is in the class AE(0)rp.



38 W. W. Comfort and A. W. Hager

(This shows that the converse to Theorem 2.1 fails: in a P -space, each
Gδ-point is isolated.)

Corollary 2.8. Let T be a space.

(a) If each x ∈ T has a neighborhood Ux ∈ AE(0)
rp, then T ∈ AE(0)rp;

and
(b) if T is the topological sum (the “disjoint union”) of spaces in AE(0)rp,

then T ∈ AE(0)rp.

Proof. It suffices to prove (a), since (b) is then immediate.
By Theorem 2.1, it suffices to show that every compact S ⊆ T is metrizable.

Let {Ux : x ∈ T } be a cover of T as indicated (with each Ux ∈ AE(0)
rp), and

for x ∈ T choose open Vx such that x ∈ Vx ⊆ Vx ⊆ Ux. There is finite F ⊆ S
such that

S ⊆
⋃

x∈F
Vx ⊆

⋃
x∈F

Vx ⊆
⋃

x∈F
Ux

and hence S =
⋃

x∈F
(S∩Vx). Each space S∩Vx is compact (being closed in S)

and is in AE(0)rp (being a subset of Ux ∈ AE(0)
rp). So by Theorem 2.2, each

space S ∩ Vx is metrizable. Thus S, the union of finitely many of its closed,
metrizable subspaces, is itself metrizable ([4](4.19)). �

3. An Application to Lattice-Ordered Groups

We consider the category W∗ of archimedean lattice-ordered groups G with
distinguished strong order unit eG (that means: for each g ∈ G there is n ∈ N
such that |g| ≤ neG), together with group- and lattice-homomorphisms which
preserve unit. The notation G ≤ H indicates that G ∈ W∗ is a subobject of
H ∈ W∗. The Yosida representation theorem, as exposed in [9], tells us that

each G ∈ W∗ has an essentially unique representation G ≃ Ĝ ≤ C(Y G, R) with

Y G compact (Hausdorff) and with Ĝ separating points of Y G; and, for each
φ : G → H ∈ W∗ there corresponds a unique continuous τ : Y H ։ Y G such

that φ̂(g) = ĝ ◦τ for each g ∈ G, and with τ an injection (hence an embedding)

if φ is a surjection. We identity each G ∈ W∗ with its Ĝ. Thus, a surjection
φ : G ։ H becomes the restriction to Y H of the functions in G.

Now let E ≤ R (that is, E is a subgroup of R, and 1 ∈ E), and set
CE := {C(X, E) : X is compact} ⊆ W

∗.

Theorem 3.1. CE is closed under surjections in W
∗.

Proof. [We sketch.] First consider the case E = R. Then Y C(X, R) = X, and
each surjection φ : C(X, R) ։ H is induced by the restriction g → g|Y H to
the subspace Y H ⊆ X. Each f ∈ C(Y H, R) has an extension g ∈ C(X, R)
(Tietze-Urysohn), so f = g|Y H and H = C(Y H, R).

Now if E 6= R then E is zero-dimensional, and Y := Y C(X, E) is the zero-
dimensional reflection of X: Y ∈ 0. So for a surjection φ : C(X, E) ։ H the
“dual” topological inclusion Y H ⊆ Y lives in 0. Then, each f ∈ C(Y H, E)
has an extension g ∈ C(Y, E), because E ∈ AE(0)rp (e.g., by Theorem 2.2),
so f = g|Y H and again H = C(Y H, E), as required. �



Range-preserving AE(0)-spaces 39

We note that when E 6= R in the preceding theorem, either E is cyclic
(and thus discrete) or E is dense in R. In the former case, an extension g of
f ∈ C(Y H, E) is easily manufactured, using the fact that |f[Y H]| < ω, by
extending the resulting finite clopen partition of Y H to one of Y (much as
in the proof of Theorem 1.1). In the (proof of the) dense case, however, the
relation E ∈ AE(0)rp is crucial; the proof of that appears to require much of
the argumentation we have given above in Theorem 2.2.

More issues of the sort addressed in this section are considered in the work
[9].

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(Received November 2011 – Accepted November 2012)



40 W. W. Comfort and A. W. Hager

W. W. Comfort (wcomfort@wesleyan.edu)
Department of Mathematics and Computer Science, Wesleyan University, Mid-
dletown, CT 06459, USA

A. W. Hager (ahager@wesleyan.edu)
Department of Mathematics and Computer Science, Wesleyan University, Mid-
dletown, CT 06459, USA


	Range-preserving AE(0)-spaces. By W. W. Comfort and A. W. Hager