@ Appl. Gen. Topol. 22, no. 1 (2021), 199-221doi:10.4995/agt.2021.14915
© AGT, UPV, 2021

On the Menger and almost Menger properties

in locales

Tilahun Bayih, Themba Dube and Oghenetega Ighedo

Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria,

South Africa (tileesa24@gmail.com,dubeta@unisa.ac.za,ighedo@unisa.ac.za)

Communicated by F. Mynard

Abstract

The Menger and the almost Menger properties are extended to locales.
Regarding the former, the extension is conservative (meaning that a
space is Menger if and only if it is Menger as a locale), and the latter
is conservative for sober TD-spaces. Non-spatial Menger (and hence
almost Menger) locales do exist, so that the extensions genuinely tran-
scend the topological notions. We also consider projectively Menger
locales, and show that, as in spaces, a locale is Menger precisely when
it is Lindelöf and projectively Menger. Transference of these properties
along localic maps (via direct image or pullback) is considered.

2010 MSC: 06D22; 54C05; 54D20.

Keywords: Menger; almost Menger; frame; locale; sublocale; spectrum of a
frame.

Introduction

Recall that a topological space X is Menger if for every sequence (Un)n∈N of
open covers of X we can select, for each n, a finite Vn ⊆ Un such that

⋃
n∈NVn

is a cover of X. This definition is purely in terms of the lattice of open subsets,
and can thus be extended to frames almost verbatim. That is exactly what
we do. It then turns that the extension of the Menger property to frames is
conservative.

On the other hand, a topological space is called almost Menger if for every
sequence (Un)n∈N of open covers of X we can select, for each n, a finite Vn ⊆ Un

Received 11 January 2021 – Accepted 11 February 2021

http://dx.doi.org/10.4995/agt.2021.14915


T. Bayih, T. Dube and O. Ighedo

such that
⋃
{V | V ∈

⋃
n∈NVn} = X. Although this definition is not solely

in terms of the lattice of open sets (because of the appearance of closures),
it can be adapted to frames by working within the lattice of sublocales, with
the union replaced by the join. That is precisely what we do to define almost
Menger frames.

Our aim in this paper is to initiate the study of the Menger-type properties
in pointfree topology. Some of the results we obtain not only extend the known
topological ones to frames, but also sharpen the topological ones. There are
various weaker forms of the Menger property in spaces, but we restrict ourselves
to extensions of the Menger property and the almost Menger property.

Here is a brief overview of the paper. Since the theory of frames and locales
has by now come of age, the preliminaries in Section 1 are written tersely; the
main purpose being just to fix notation and recall the concepts that are used
most throughout the paper.

In Section 2 we study some properties of Menger frames. We start by ob-
serving that (as already been mentioned) a topological space X is Menger if
and only if the frame Ω(X) is Menger, and that non-spatial Menger frames do
exist, so that our extension to frames of this property is a genuine extension
covering more objects than topologies of Menger spaces.

Since the contravariant functor Ω: Top → Frm preserves and reflects the
Menger property, one may ask about its right adjoint Σ: Frm → Top. A
frame whose spatial reflection is a codense sublocale is Menger if and only if its
spectrum is Menger (Proposition 2.6). It is perhaps worth underscoring that a
frame whose spatial reflection is a codense sublocale is not necessarily spatial.

Defining a frame to be projectively Menger if every subframe with a count-
able base is Menger, we have that a frame is Menger precisely when it is Lin-
delöf and projectively Menger (Corollary 2.14). A completely regular normal
countably paracompact frame is projectively Menger if and only if its Lindelöf
coreflection is Menger (Corollary 2.16).

In Section 3 we consider almost Menger frames. Our definition, adapted
from spaces as indicated above, turns out to be conservative for sober TD-spaces
(Theorem 3.3). Although our definition invokes the lattice of sublocales, we
have a characterisation (Proposition 3.6) solely in terms of elements.

1. Preliminaries

We assume familiarity with frames and locales. Our references are [12] and
[15]. In this section we recall just a few of the concepts that we shall need. Our
notation is standard, and is, by and large, that of our references.

1.1. Frames and spatiality. Throughout this section, L denotes a frame.
We denote by Ω(X) the frame of open subsets of a topological space X. An
element p ∈ L is called a point (or a prime) if it satisfies the property that

p < 1 and (∀x, y ∈ L)(x ∧ y ≤ p =⇒ x ≤ p or y ≤ p).

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On the Menger and almost Menger properties in locales

We write Pt(L) for the set of points of L. A frame is spatial if it is isomorphic
to Ω(X) for some space X. This is the case precisely when every element is a
meet of primes.

We view the spectrum of L as the topological space whose underlying set is
Pt(L) with the topology

Ω(ΣL) = {Σa | a ∈ L} where, for each a ∈ L, Σa = {p ∈ Pt(L) | a � p}.

The map ηL : L → Ω(ΣL) given by ηL(a) = Σa is an onto frame homomor-
phism, and is the reflection map from L to spatial frames.

As usual, we shall write ≺ and ≺≺, respectively, for the rather below and the
completely below relations, and recall that L is called regular (resp. completely
regular) if every element of L is the join of the elements that are rather below
(resp. completely below) it.

1.2. Sublocales and localic maps. The lattice of sublocales of L, ordered
by inclusion, is a coframe denoted by S(L). For later use, we recall that joins
in S(L) are given by

∨

i∈I

Si =
{∧

M | M ⊆
⋃

i∈I

Si

}
.

Since S(L) is a coframe, when turned upside down it is a frame, denoted S(L)op,
whose top element is the void sublocale O = {1}. A sublocale of L is called
a one-point sublocale if it is of the form {p, 1} for some p ∈ Pt(L). Spatial
frames are precisely those that are joins of their one-point sublocales.

The open sublocale associated with a ∈ L is denoted by oL(a), and the closed
one by cL(a). We shall drop the subscript if no confusion may result from that.
The closure of a sublocale S of L, denoted S or clL S, is the sublocale

S = cL

(∧
S
)
.

In particular, oL(a) = cL(a
∗). A sublocale S of L is dense if S = L. If S and

T are sublocales of L and S ⊆ T , then S is a sublocale of T . The closure of S
in T will be denoted by clT S, and S (unadorned) will be understood to be the
closure in L.

A localic map f : L → M gives rise to two maps

f[−]: S(L) → S(M) and f−1[−]: S(M) → S(L)

given by

f[S] = {f(x) | x ∈ S} and f−1[T ] =
∨

{A ∈ S(L) | A ⊆ f−1[T ]}.

The map f[−] preserves all joins and f−1[−] preserves all meets (recall that
they are intersections) and all binary joins. For any S ∈ S(L) and T ∈ S(M),

f[S] ⊆ T ⇐⇒ S ⊆ f−1[T ].

Writing h for the left adjoint of f, we have that, for any b ∈ M,

f−1[oM(b)] = oL(h(b)) and f−1[cM (b)] = cL(h(b)).

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T. Bayih, T. Dube and O. Ighedo

This then shows that the map f−1[−] also preserves arbitrary joins of open
sublocales. For, if {bi | i ∈ I} ⊆ M, then

f−1

[∨

i∈I

oM(bi)
]
= f−1

[
oM

(∨

i∈I

bi

)]
= oL

(
h
(∨

i∈I

bi

))

= oL

(∨

i∈I

h(bi)
)
=
∨

i∈I

oL
(
h(bi)

)
=
∨

i∈I

f−1[oM (bi)].

1.3. Covers and coverings. By a cover of L we mean a set C ⊆ L such that∨
C = 1. On the other hand, to avoid possible confusion, we say a collection

C of sublocales of L is a covering of L if
∨
{C | C ∈ C } = L, where the join

is calculated in S(L). This terminology is not standard. A cover consists of
elements of L, whereas a covering consists of sublocales of L. If every sublocale
in a covering C of L is open, then C is an open covering of L. There is a
bijection between covers and open coverings given by

C 7→ C C = {oL(c) | c ∈ C} and C 7→ C
C = {x ∈ L | oL(x) ∈ C }.

A cover C of L is said to refine a cover D if for every c ∈ C there is a d ∈ D
such that c ≤ d. In this case, C is called a refinement of D.

2. Menger locales

We aim to define Menger locales in such a way that a space X is Menger
precisely when the frame Ω(X) is Menger. Our definition will be localic, and
we will then cast it in frame terms, which will enable us to show easier that
the definition is conservative. Throughout, every sequence is indexed by N.

Definition 2.1. A frame L is Menger if for every sequence (Cn) of open
coverings of L, there exists, for each n, a finite Dn ⊆ Cn such that

⋃
n∈NDn is

a covering of L. In this case, we say the sequence (Dn) is a Menger witness for
(Cn).

From the bijection between covers and coverings, this definition could equiv-
alently have been stated in terms of covers. The reason is that if C is a cover of
L and D is a finite subset of C, then, in the notation of Subsection 1.3, C D is a
finite subset of C C. Conversely, if C is an open covering of L and D is a finite
subset of C , then CD is a finite subset of CC because the mapping u 7→ oL(u)
is one-one.

Proposition 2.2. A frame L is Menger iff for every sequence (Cn) of covers
of L, there exists, for each n, a finite Dn ⊆ Cn such that

⋃
n∈NDn is a cover

of L.

As with coverings, we shall say such a sequence (Dn) is a Menger witness
for the sequence (Cn). This proposition makes it most apparent that every
Menger frame is Lindelöf, and every compact frame (in fact, every σ-compact
one – meaning one that is a join of countably many compact sublocales) is
Menger. Since there are non-spatial compact frames (see [12, p. 89]), it follows
that:

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On the Menger and almost Menger properties in locales

A Menger frame need not be spatial.

Since every cover of a subframe is a cover of the ambient frame, we deduce
that:

Every subframe of a Menger frame is Menger. Thus, a localic

image of a Menger frame is Menger.

Since a collection of open subsets of a space X is a cover of the frame Ω(X) if
and only if it is an open cover of the space X, we deduce the following from
Proposition 2.2.

Corollary 2.3. A topological space X is Menger iff Ω(X) is Menger.

Recall that the sobrification of a topological space is the spectrum of its
frame of open sets. Since a space and its sobrification have isomorphic frames
of open sets, we have the following result.

Corollary 2.4. A topological space is Menger iff its sobrification is Menger.

In light of the dual adjunction

Top
Ω

//

Frm
Σ

oo

and the result in Corollary 2.3, one may ask if it is the case that a frame is
Menger if and only if its spectrum is Menger. We address this for some types
of frames. As is well known, a frame L is spatial if and only if the frame
homomorphism ηL : L → Ω(ΣL) is one-one. We will show that for frames L
for which ηL is codense (meaning that ηL(a) = 1Ω(ΣL) implies a = 1L) the
spectrum analogue of Corollary 2.3 holds. We reiterate that such frames need
not be spatial, as the following example shows.

Example 2.5. Let L be a frame with no points, such as the smallest dense
sublocale of Ω(R). Let L̃ be the frame obtained from L by adjoining a new top
element 1L̃ > 1L. Then L̃ is not spatial and Pt(L̃) = {1L}. From the latter, it
is not hard to see that ηL̃ is codense.

Proposition 2.6. A frame whose spatial reflection is a codense sublocale is

Menger iff its spectrum is Menger.

Proof. Let L be such a frame. For any A ⊆ L we set ΣA = {Σa | a ∈ A}. Since
Σ∨

i∈I
ai =

⋃
i∈IΣai for any collection {ai | i ∈ I} of elements of L, it follows

that ΣC is an open cover of ΣL whenever C is a cover of L. On the other hand,
the part of the hypothesis that says ηL is codense ensures that every open cover
of ΣL is of the form ΣC for some cover C of L.

Now assume that ΣL is Menger. We apply Proposition 2.2 to show that L
is Menger. Let (Cn) be a sequence of covers of L. Then (ΣCn) is a sequence
of open covers of ΣL. So for each n there exists a finite Dn ⊆ Cn such that⋃

n∈NΣDn = ΣL. Since
⋃

n∈NΣDn = Σ
⋃

n∈N
Dn, we deduce from the codensity

part of the hypothesis that
⋃

n∈NDn is a cover of L. Therefore L is Menger.
The converse is shown similarly. �

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T. Bayih, T. Dube and O. Ighedo

When working with covers or coverings, it is at times convenient to deal
with directed ones. When we say a subset of a poset is directed, we mean that
it is up-directed. If C is a cover of L and C is an open covering of L, we set

C× =
{∨

F | F is a finite subset of C
}

and

C
× =

{∨
F | F is a finite subset of C

}
,

and observe that C× is a directed cover of L and C × is directed (open) covering
of L.

Proposition 2.7. The following are equivalent for a frame L.

(1) L is Menger.
(2) For every sequence (Cn) of directed covers of L, there exists, for each

n, an element cn ∈ Cn such that {cn | n ∈ N} is a cover of L.
(3) For every sequence (Un) of directed open coverings of L, there exists,

for each n, a sublocale Un ∈ Un such that {Un | n ∈ N} is a covering
of L.

Proof. (2) ⇔ (3): This equivalence comes from the bijection between covers
and open coverings, together with the observation that this bijection induces a
bijection between directed covers and directed open coverings.

(1) ⇔ (2): Assume that L is Menger, and let (Cn) be a sequence of directed
covers of L. Let (Dn) be a Menger witness for (Cn). Since each Dn is a finite
subset of Cn and Cn is directed, there exists an element cn ∈ Cn such that∨
Dn ≤ cn. Therefore the cover

⋃
n∈NDn refines {cn | n ∈ N}, showing that

the latter is a cover of L.
Conversely, let (Cn) be a sequence of covers of L, and consider the sequence

(C×n ) of directed covers of L. The current hypothesis furnishes, for each n, a
finite Bn ⊆ Cn such that the set {

∨
Bn | n ∈ N} is a cover of L. Clearly, this

makes the sequence (Bn) a Menger witness for (Cn), and hence L is a Menger
frame. �

Now, let us observe that the Menger property is preserved under finite joins.
In the proof we shall use the fact that families of open sublocales are distributive
in the coframe of sublocales, that is, if (Uα)α∈A is a family of open sublocales
of L and T is any sublocale of L, then

T ∩
∨

α∈A

Uα =
∨

α∈A

(T ∩ Uα).

Recall that open sublocales of a sublocale A of a frame L are precisely the
intersections with A of the open sublocales of L.

Proposition 2.8. The join of finitely many Menger sublocales of a given frame

is Menger.

Proof. Let us first show that a sublocale A of L is Menger if and only if whenever
(Un) is a sequence of families of open sublocales of L such that A ⊆

∨
Un for

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On the Menger and almost Menger properties in locales

every n, then there exists, for each n, a finite Vn ⊆ Un such that A ⊆
∨⋃

n∈NVn.
To see this, let us introduce the ad hoc notation that if F is a family of
sublocales of L, we write

A ∩ F = {A ∩ F | F ∈ F }.

Now, for the “if” part, the equality A =
∨
{A ∩ U | U ∈ Un} implies that

A ∩ Un is an open covering of A for each n. Since A is Menger, we can find,
for each n, a finite Vn ⊆ Un such that A ∩

⋃
n∈NVn is a covering of A. Thus,

A =
∨{

A ∩ S | S ∈
⋃

n∈N

Vn

}
= A ∩

∨{
S | S ∈

⋃

n∈N

Vn

}
,

which implies A ⊆
∨⋃

n∈NVn, as desired. The “only if” part is proved similarly,
taking into account the fact every open covering of A is of the form A ∩ U ,
where U is a family of open sublocales of L with A ⊆

∨
U .

Now let A and B be Menger sublocales of L, and consider a sequence (Cn) of
families of open sublocales of L with A ∨ B ⊆

∨
Cn for each n. Then A ⊆

∨
Cn

and B ⊆
∨

Cn for each n. So we can find a finite D
A
n ⊆ Cn and a finite

D
B
n ⊆ Cn such that

A ⊆
∨⋃

n∈N

D
A
n and B ⊆

∨⋃

n∈N

D
B
n .

Consequently, for each n, DAn ∪ D
B
n is a finite subset of Cn such that

A ∨ B ⊆
∨{

S | S ∈ DAn ∪ D
B
n

}
,

which proves that A ∨ B is Menger. The general case follows by induction
because the binary join is an associative operation on S(L). �

The sublocales that inherit the Menger property include the closed ones
because for any frame L and a ∈ L, a cover of c(a) is a cover of L. In fact,
we have a stronger result. Recall that a frame homomorphism is called perfect
if its right adjoint preserves directed joins. In [8], a frame homomorphism is
called weakly perfect if its right adjoint preserves directed covers.

Perfect homomorphisms are weakly perfect. Weak perfectness is strictly
weaker than perfectness. Indeed, as observed in [7, Example 3.11], if L is a
compact frame which is not Boolean, then the right adjoint of the join map
JL → L (where JL denotes the frame of ideals of L) takes covers to covers
(and hence is weakly perfect), but it is not perfect. On the other hand, weak
perfectness does not imply that the right adjoint takes covers to covers. A
counterexample (also sourced from [7]) is the embedding of the two-element
chain in the four-element Boolean algebra.

We can summarise these “pictorially” using the acronyms that a frame ho-
momorphism satisfies:

• (DJ) if its right adjoint preserves directed joins;
• (DC) if its right adjoint sends directed covers to covers; and
• (CC) if its right adjoint sends covers to covers.

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T. Bayih, T. Dube and O. Ighedo

Then
(DJ) 6=⇒ (CC) 6=⇒ (DJ); (CC) =⇒ (DC) 6=⇒ (CC)

and
(DJ) =⇒ (DC) 6=⇒ (DJ).

Thus, weak perfectness is the least restricted of these properties of homomor-
phisms. So a property of frames that is preserved or reflected by weakly perfect
homomorphisms is also done so by the other types of homomorphisms.

In [6], a cover B of a frame L is called a strong refinement of a cover C if
for every b ∈ B there is a c ∈ C such that b ≺ c. Then L is called cover regular
if every cover of L has a strong refinement. Regular frames are cover regular.
Any finite chain with more than two elements is a cover regular frame which
is not regular. Note that if a directed cover has a strong refinement, then it
has a strong refinement which is directed because whenever bi ≺ ci for each
i ∈ {1, . . . , n}, with n ∈ N, then (b1 ∨ · · · ∨ bn) ≺ (c1 ∨ · · · ∨ cn).

Recall that a frame homomorphism h: L → M is called dense if the zero
of its domain is the only element mapped to the zero of its codomain. This is
precisely when h∗(0) = 0, where h∗ denotes the right adjoint of h. It is well
known that if h is a dense frame homomorphism, then h∗(h(x)) ≤ a whenever
x ≺ a in the domain of h.

Corollary 2.9. Let h: L → M be a frame homomorphism.

(a) If h is weakly perfect and L is Menger, then M is Menger.
(b) If h is dense and weakly perfect, L is cover regular, and M is Menger,

then L is Menger.

Proof. (a) Let (Cn) be a sequence of directed covers of M. Since h is weakly
perfect, h∗[Cn] is a directed cover of L for each n. Since L is Menger, by
Proposition 2.7 there exists, for each n, an element cn ∈ Cn such that {h∗(cn) |
n ∈ N} is a cover of L. Therefore {hh∗(cn) | n ∈ N} is a cover of M, and
hence {cn | n ∈ N} is a cover of M because hh∗(x) ≤ x for every x ∈ M.
Proposition 2.7 again shows that M is Menger.

(b) Let (Cn) be a sequence of directed covers of L, and, for each n, let Bn
be a directed strong refinement of Cn. Then h[Bn] is a directed cover of M
for each n. Since M is Menger, Proposition 2.7 enables us to find, for each
n ∈ N, an element bn ∈ Bn such that {h(bn) | n ∈ N} is a cover of M. Since h∗
takes directed covers to covers, the set C = {h∗h(bn) | n ∈ N} is a cover of L.
Since Bn is a strong refinement of Cn, for each n, there exists some cn ∈ Cn
such that bn ≺ cn, and since h is dense, we have h∗h(bn) ≤ cn. So the cover C
refines {cn | n ∈ N}, whence we deduce that L is Menger. �

Part (a) of this corollary enables us to say a word about coproducts. We do
not need to recall the construction of coproducts.

Corollary 2.10. If L is compact and M is Menger, then L ⊕ M is Menger.

Proof. It is shown in [11, Lemma 2] that if L is compact, then the coproduct
injection M → L ⊕ M is a perfect map (actually, it is a proper map – we

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On the Menger and almost Menger properties in locales

will recall the definition later). Since any perfect homomorphism is weakly
perfect, the result follows from Corollary 2.9(a) because compact frames are
Menger. �

We recall that a topological space X is called “projectively Menger” if every
continuous second countable image of X is Menger. In [4], it is shown that a
Tychonoff space is Menger if and only if it is Lindelöf and projectively Menger.
We extend this result to frames without restricting to completely regular ones.
Let us first recall some frame-theoretic notions. A subset S of a frame L is said
to be a generating set if L is the smallest (under inclusion) subframe containing
S. This is the case precisely when each a ∈ L is of the form

a =
∨

{x ∈ L | x is the meet of some finite F ⊆ S}.

A base of a frame L is a subset B with the property that every element of L is
a join of some elements of B.

Definition 2.11. A frame L is projectively Menger if every subframe of L with
a countable base is Menger.

This definition can clearly be rephrased to say L is projectively Menger
in case whenever h: M → L is a one-one frame homomorphism and M has a
countable base, then M is Menger. Let us record the following easy observation,
which should certainly be known, but for which we provide a proof as we do
not have a reference.

Lemma 2.12. Every frame with a countable base is Lindelöf.

Proof. Let B be a countable base of a frame L. Let C be a cover of L. For
each c ∈ C, let B(c) be a subset of B such that c =

∨
B(c). Then

⋃
c∈CB

(c) is
a countable cover of L refining C. Therefore C has a countable subcover. �

In what follows, we say a countable cover C = {c1, c2, . . . } is increasing if
cn ≤ cn+1 for each n. This is standard terminology. Of course, an increasing
cover is directed. For any given countable cover C = {c1, c2, . . . }, we denote
by

C+ = {c1, c1 ∨ c2, c1 ∨ c2 ∨ c3, . . . }

the increasing cover constructed from C as indicated.

Proposition 2.13. The following are equivalent for a frame L.

(1) L is projectively Menger.
(2) Every Lindelöf subframe of L is Menger.
(3) For every sequence (Cn) of countable covers of L, there exists, for each

n, a finite Dn ⊆ Cn such that
⋃

n∈NDn is a cover of L.
(4) For every sequence (Cn) of increasing countable covers of L, there ex-

ists, for each n, an element cn ∈ Cn such that {cn | n ∈ N} is a cover
of L.

(5) For every sequence (Cn) of countable open coverings of L, there exists,
for each n, a finite Dn ⊆ Cn such that

⋃
n∈NDn is a covering of L.

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T. Bayih, T. Dube and O. Ighedo

Proof. (1) ⇒ (4): Assume that L is projectively Menger, and let (Cn) be a
sequence of increasing countable covers of L. Put C =

⋃
n∈NCn. Then C is

a countable subset of L. Let M be the subframe of L generated by C. Then
the set CF whose elements are the meets of finite subsets of C is a base for
M. Since C is countable, CF is countable, and so M has a countable base.
Therefore M is Menger, by hypothesis. Since each Cn ⊆ M, (Cn) is a sequence
of increasing covers of M, so for each n there exists some cn ∈ Cn such that
{cn | n ∈ N} is a cover of M, and hence of L. Therefore (1) implies (4).

(4) ⇒ (3): Assume that condition (4) holds, and let (Cn) be a sequence of
countable covers of L. Consider the sequence (C+n ) of increasing covers of L.
By (4), there exists, for each n, some dn ∈ C

+
n such that {dn | n ∈ N} is a

cover of L. Since each dn is a join of some finite Dn ⊆ Cn,
⋃

n∈NDn is a cover
of L.

(3) ⇒ (2): Assume condition (3), and let M be a Lindelöf subframe of L.
Let (Un) be a sequence of covers of M. Since M is Lindelöf, we can find, for
each n, a countable Cn ⊆ Un such that Cn is a cover of M. Then of course
Cn is a cover of L, and so, by (3), there exists a finite Dn ⊆ Cn such that⋃

n∈NDn is a cover of L, and hence of M. Thus, M is Menger, and therefore
L is projectively Menger.

(2) ⇒ (1): This follows from the fact that every frame with a countable base
is Lindelöf.

(4) ⇔ (5): This follows from the bijection between covers and open cover-
ings. �

Since every Menger frame is Lindelöf and every subframe of a Menger frame
is Menger, we deduce the following characterisation of Menger frames from this
proposition.

Corollary 2.14. A frame is Menger iff it is Lindelöf and projectively Menger.

We now turn to some subclass of completely regular frames. Recall from [6]
that a frame L is countably paracompact if for every countable cover C of L,
there is a cover W of L such that each element of W misses all but a finite
number of elements of C. Let us also recall that a countable cover D = {dn |
n ∈ N} is called a shrinking of a countable cover C = {cn | n ∈ N} if dn ≺ cn
for every n ∈ N. By a cozero cover of a frame we mean a cover consisting
entirely of cozero elements.

Theorem 2.15. The following are equivalent for a normal countably paracom-

pact completely regular frame L.

(1) L is projectively Menger.
(2) For every sequence (Cn) of countable cozero covers of L, there exists,

for each n, a finite Dn ⊆ Cn such that
⋃

n∈NDn is a cover of L.
(3) For every sequence (Cn) of increasing countable cozero covers of L,

there exists, for each n, an element cn ∈ Cn such that {cn | n ∈ N} is
a cover of L.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 208



On the Menger and almost Menger properties in locales

Proof. The implication (1) ⇒ (2) follows from Proposition 2.13, and the impli-
cation (2) ⇒ (3) is almost immediate.

(3) ⇒ (1): Let (Un) be a sequence of increasing countable covers of L. Now
fix n ∈ N, and write

Un = {un1, un2, . . . } with un1 ≤ un2 ≤ · · · .

Since L is countably paracompact, Un has a shrinking [5, Corollary to Propo-
sition 1]. Since x ∨ y ≺ a whenever x ≺ a and y ≺ a, and since x ≤ a ≺ b
implies x ≺ b, the cover Un actually has an increasing shrinking

Vn = {vn1, vn2, . . . } with vnk ≺ unk for each k ∈ N.

By normality, vnk ≺≺ unk for each k, and hence, by [2, Corollary 1], there is a
cozero element znk such that vnk ≺≺ znk ≺≺ unk. Since the join of finitely many
(actually, countably many) cozero elements is a cozero element, the sequence
(znk)k∈N can be chosen so that it is increasing. Thus, if for each n ∈ N we let
Zn be the set

Zn = {znk | k ∈ N},

then (Zn) is a sequence of countable increasing cozero covers of L, and so by (3)
there exists, for each n, an element zn ∈ Zn such that {zn | n ∈ N} is a cover
of L. Since each Zn refines Un, there exists, for each n, an element un ∈ Un
such that {un | n ∈ N} is cover of L. Therefore L is projectively Menger by
Proposition 2.13. �

Let L be a completely regular frame and λL be its Lindelöf coreflection
(see [14] for details). As shown in [1, Corollary 8.2.13], if we let h: λL → L
be the coreflection map to L from completely regular Lindelöf frames, then for
any countable cozero cover C of L, h∗[C] is a (countable) cozero cover of λL.
Since, as is well known, Coz(λL) = {h∗(c) | c ∈ Coz L}, the countable cozero
covers of λL are precisely the covers h∗[C], for C a countable cozero cover of
L. Furthermore, h∗[C] is increasing if and only if C is increasing. In all, this
yields the following corollary.

Corollary 2.16. A normal countably paracompact completely regular frame L
is projectively Menger iff λL is Menger.

Proof. Suppose that L is projectively Menger, and denote by h: λL → L the
coreflection map to L from completely regular Lindelöf frames. Let (h∗[Cn])
be a sequence of increasing countable cozero covers of λL. Then (Cn) is a
sequence of increasing countable cozero covers of L. By Theorem 2.15, there
exists, for each n, an element cn ∈ Cn such that {cn | n ∈ N} is a cover of L.
Thus, for each n, there exists dn ∈ h∗[Cn] such that {dn | n ∈ N} is a cover
of λL. Therefore λL is projectively Menger by Theorem 2.15, and hence it is
Menger by Corollary 2.14 because it is Lindelöf.

The converse is proved similarly. �

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 209



T. Bayih, T. Dube and O. Ighedo

3. Almost Menger frames

Apart from the projective Menger property that we discussed in the previous
section (solely for purposes of characterising Menger frames), we shall also
consider a weaker form of the Menger property in frames. As with the case of
the Menger property, we will define it by a condition lifted straight from spaces,
mutatis mutandis. We will show that the localic extension is conservative
for sober TD-spaces. Recall that a space is sober if it is a T0-space and the
complements of the closures of its singletons are exactly its meet-irreducible
open sets. That is, a space X is sober if and only if it is a T0-space and

Pt(Ω(X)) = {X r {x} | x ∈ X}.

On the other hand, X is a TD-space if each x ∈ X has an open neighbourhood
U such that U r {x} is open. Hausdorff spaces are sober TD-spaces.

Let us recall that a space X is called almost Menger if for every sequence
(Cn) of open covers of X, there exists, for each n, a finite Dn ⊆ Cn such that⋃{

D | D ∈
⋃

n∈NDn

}
= X.

Definition 3.1. A frame L is almost Menger if for every sequence (Cn) of
open coverings of L, there exists, for each n, a finite Dn ⊆ Cn such that∨
{D | D ∈

⋃
n∈NDn

}
= L. In this case, we say the sequence (Dn) is an almost

Menger witness for the sequence (Cn).

It is clear that every Menger frame is almost Menger. Before investigating
some of the properties of almost Menger frames, we show that, among sober
TD-spaces, the definition of almost Menger frames is conservative. We need
some background. Recall (from [16], for instance) that a prime element p of a
frame L is said to be a covered prime if whenever p =

∧
S for some S ⊆ L, then

p ∈ S. Since for any sober space X the prime elements of Ω(X) are precisely

the open sets X r {x}, for x ∈ X, we deduce from [16, Proposition 1.6.2] that
if X is a sober TD-space, then all its prime elements are covered.

We recalled in the Preliminaries that

L is spatial iff L =
∨{

{p, 1} | p ∈ Pt(L)
}
.

If X is sober, then the one-point sublocales of Ω(X) are exactly the sublocales

{X r {x}, 1Ω(X)}, with x ∈ X. Hence, if X is sober, then

Ω(X) =
∨{

{X r {x}, 1Ω(X)} | x ∈ X
}
.

In fact, this equality holds without sobriety, as observed in [17]. The argument
goes as follows. If X is any topological space and U ⊆ X is open, then U =⋂{

X r {w} | w /∈ U
}
, so that, calculating in Ω(X) and S(Ω(X)),

U = int

(
⋂

w /∈U

(
X r {w}

)
)

=
∧

w /∈U

(
X r {w}

)
∈
∨{

{X r {x}, 1Ω(X)} | x ∈ X
}
.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 210



On the Menger and almost Menger properties in locales

Lemma 3.2. Let {Uα | α ∈ A} be a family of open subsets of a topological
space X.

(a) If
⋃

α∈AUα = X, then
∨{

oΩ(X)(Uα) | α ∈ A
}
= Ω(X).

(b) If X is a sober TD-space and
∨{

oΩ(X)(Uα) | α ∈ A
}
= Ω(X), then⋃

α∈AUα = X.

Proof. (a) Note that

oΩ(X)(Uα) = cΩ(X)(U
∗
α) = cΩ(X)(X r Uα).

Now, given p ∈ X, since
⋃

α∈AUα = X, there is an index γ ∈ A such that

p ∈ Uγ, which implies {p} ⊆ Uγ, and hence X r Uγ ⊆ X r {p}. Consequently,
the element X r{p} of the frame Ω(X) belongs to the sublocale cΩ(X)(X rUγ),
whence we deduce that

{X r {p}, 1Ω(X)} ⊆ cΩ(X)(X r Uγ)

and therefore,

Ω(X) =
∨{

{X r {x}, 1Ω(X)} | x ∈ X
}

⊆
∨{

cΩ(X)(X r Uα) | α ∈ A
}

=
∨{

oΩ(X)(Uα) | α ∈ A
}

⊆ Ω(X),

which proves the claimed equality.

(b) Assume that X is a sober TD-space and
∨{

oΩ(X)(Uα) | α ∈ A
}
= Ω(X).

Then, for any x ∈ X, the element X r{x} of Ω(X) belongs to this join, so that

X r {x} ∈
∨{

cΩ(X)(X r Uα) | α ∈ A
}
.

By the way joins of sublocales are computed, for each α there is an open subset
Vα of X such that X r Uα ⊆ Vα and X r {x} =

∧
αVα. Since primes are

covered here, there is an index γ ∈ A such that X r {x} = Vγ. Therefore
x ∈ X r Vγ ⊆ Uγ, which then shows that

⋃
α∈AUα = X. �

Theorem 3.3. Let X be a topological space.

(a) If X is almost Menger, then Ω(X) is almost Menger.
(b) If X is a sober TD-space, then Ω(X) is almost Menger iff X is almost

Menger.

Proof. (a) Let (Cn) be a sequence of open coverings of Ω(X). For each n,
there is an open cover Un of X such that Cn = {oΩ(X)(U) | U ∈ Un}. Now,
since X is almost Menger, there exists, for each n, a finite Vn ⊆ Un such that⋃
{V | V ∈

⋃
n∈NVn} = X. For each n, put

Dn = {oΩ(X)(V ) | V ∈ Vn}.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 211



T. Bayih, T. Dube and O. Ighedo

We will show that (Dn) is an almost Menger witness for (Cn). Clearly, each
Dn is finite and Dn ⊆ Cn. Let us check the final condition. Since

⋃
{V | V ∈⋃

n∈NVn} = X, we deduce from Lemma 3.2(a) that

Ω(X) =
∨{

oΩ(X)(V ) | V ∈
⋃

n∈N

Vn

}
=
∨{

D | D ∈
⋃

n∈N

Dn

}
,

which shows that the final condition to make (Dn) an almost Menger witness
for (Cn) is satisfied. Therefore Ω(X) is almost Menger.

(b) Assume that X is a sober TD-space and the frame Ω(X) is almost
Menger. Let (Un) be a sequence of open covers of X. For each n ∈ N, put

U
′
n = {oΩ(X)(U) | U ∈ Un},

and observe that U ′n is an open covering of Ω(X); and so we have a sequence
(U ′n) of open coverings of the almost Menger frame Ω(X). In accordance with
the definition, for each n, there is a finite V ′n ⊆ U

′
n such that the sequence (V

′
n)

is an almost Menger witness for (U ′n), and hence

(†)
∨{

S | S ∈
⋃

n∈N

V
′
n

}
= Ω(X).

For each n, put

Vn = {V ∈ Ω(X) | oΩ(X)(V ) ∈ V
′
n}.

Since the mapping oΩ(X) : Ω(X) → S(Ω(X)) is injective and V
′
n is a finite set,

it follows that Vn is a finite set. Furthermore, Vn ⊆ Un because if V ∈ Vn,
then oΩ(X)(V ) ∈ V

′
n ⊆ U

′
n, so that oΩ(X)(V ) = oΩ(X)(U) for some U ∈ Un,

whence V = U. Observe that, for any sublocale T of L,

T ∈
⋃

n∈N

V
′
n ⇐⇒ T = oΩ(X)(V ) for some V ∈

⋃

n∈N

Vn,

and so, by Lemma 3.2(b), the equality in (†) implies
⋃{

V | V ∈
⋃

n∈N

Vn

}
= X,

which then shows that X is almost Menger. �

Next, we show that in the definition of almost Menger frames, “open cov-
erings” can be replaced with “directed open coverings” without violating the
concept. In the proof we are going to use the fact that

S1 ∨ · · · ∨ Sn = S1 ∨ · · · ∨ Sn

for any collection of finitely many sublocales [15, Proposition III.8.1].

Proposition 3.4. A frame L is almost Menger iff for every sequence (Cn) of
directed open coverings of L, there exists, for each n, a sublocale Cn ∈ Cn such
that

∨
{Cn | n ∈ N} = L.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 212



On the Menger and almost Menger properties in locales

Proof. Suppose, first, that L is almost Menger. Let (Cn) be a sequence of
directed open coverings of L, and let (Dn) be an almost Menger witness for
(Cn). Since each Dn is a finite subset of Cn and Cn is directed, there exists
some Cn ∈ Cn such that D ⊆ Cn for every D ∈ Dn. It follows therefore that

L =
∨{

D | D ∈
⋃

n∈N

Dn} ⊆
∨{

Cn | n ∈ N
}
⊆ L,

which proves the left-to-right implication.
Conversely, suppose that the stated condition holds, and let (Cn) be a se-

quence of open coverings of L. For each n, consider the collection

C
×
n =

{∨
F | F is a finite subset of Cn

}
.

Then (C ×n ) is a sequence of directed open coverings of L. By hypothesis, there
exists, for each n, some Wn ∈ C

×
n such that

∨
{Wn | n ∈ N} = L.

Now, for each n ∈ N, there exists some k(n) ∈ N and elements F
(1)
n , . . . , F

k(n)
n

of Cn such that

Wn = F
(1)
n ∨ · · · ∨ F

k(n)
n and hence Wn = F

(1)
n ∨ · · · ∨ F

k(n)
n .

Consequently, if for each n we let C ′n =
{
F

(1)
n , . . . , F

k(n)
n

}
, then each C ′n is a

finite subset of Cn such that

L =
∨{

Wn | n ∈ N
}
⊆
∨{

T | T ∈
⋃

n∈N

C
′
n

}
⊆ L,

which then shows that (C ′n) is an almost Menger witness for (Cn), and hence
L is almost Menger. �

Although we could have proved directly from the definition that a localic
image of an almost Menger frame is almost Menger, we shall use this result. If
f : L → M is a localic map and U is a collection of sublocales of M, we write
f−1[U ] for the set {f−1[U] | U ∈ U }.

Proposition 3.5. A localic image of any almost Menger frame is itself almost

Menger.

Proof. Let f : L → M be an onto localic map with L almost Menger. Let (Un)
be a sequence of directed open coverings of M. Since f−1[−] preserves joins of
open sublocales, (f−1[Un])n∈N is a sequence of open coverings of L, and since
f−1[−] preserves order, this sequence is directed. Since L is almost Menger,
the foregoing proposition furnishes, for each n, an element Un ∈ Un such that∨
{f−1[Un] | n ∈ N} = L. Since f[−] preserves joins and f[−]◦f−1[−] ≤ idS(M),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 213



T. Bayih, T. Dube and O. Ighedo

we have (in light of f being onto)

M = f[L] = f
[∨

n∈N

f−1[Un]
]
=
∨

n∈N

f
[
f−1[Un]

]

⊆
∨

n∈N

f
[
f−1
[
Un
]]

⊆
∨

n∈N

Un ⊆ M,

which shows that L is almost Menger by Proposition 3.4. �

As with the case of Menger frames, the almost Menger ones can also be
characterised frame-theoretically without invoking sublocales. To do this, given
a collection {oL(aα) | α ∈ A} of open sublocales of L, we note that

∨

α∈A

oL(aα) = L ⇐⇒
∨

α∈A

cL(a
∗
α) = L

⇐⇒ (∀a ∈ L)
(
a =

∧

α∈A

tα, for some tα ≥ a
∗
α

)
;

where we have surreptitiously used the fact that

L =
{∧

M | M ⊆
⋃

a∈A

cL(a
∗
α)
}

and that if M ⊆ L is expressible as a union M =
⋃

i∈IMi of some subsets,
then, setting mi =

∧
Mi for each i, we have

∧
M =

∧

i∈I

mi.

Given a sequence (Cn) of covers of L, suppose that, for each n, there is a
finite Dn ⊆ Cn such that every element a of L is expressible as a =

∧
αtα

where each tα ≥ d
∗
α for some dα ∈

⋃
n∈NDn. We then say the sequence (Dn) is

an almost Menger witness for the sequence (Cn). Recall the bijection between
covers and coverings. The calculation above shows that a sequence of covers
has an almost Menger witness if and only if the corresponding sequence of open
coverings has an almost Menger witness. Consequently we have the following
result.

Proposition 3.6. A frame L is almost Menger iff for every sequence (Cn) of
covers of L, there exists, for each n, a finite Dn ⊆ Cn such that every element
a of L is expressible as a =

∧
αtα where each tα ≥ d

∗
α for some dα ∈

⋃
n∈NDn.

We have seen that working with directed covers (or coverings) is often neater.
After all, informally speaking, selecting an element is easier and quicker than
selecting a finite subset. The frame-theoretic characterisation just stated can
be couched in terms of directed covers.

Corollary 3.7. A frame L is almost Menger iff for every sequence (Cn) of
directed covers of L, we can select, for each n, an element cn ∈ Cn such that
any a ∈ L is expressible as a =

∧
n∈Ntn for some elements tn ∈ L with each

tn ≥ c
∗
n.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 214



On the Menger and almost Menger properties in locales

Proof. This follows from Proposition 3.6, the equivalences displayed in the
paragraph following the proof of Proposition 3.5, and the fact that if (Cn) is a
sequence of directed covers of L, then (C Cn)n∈N is a sequence of directed open
coverings of L, and, conversely, if (Cn) is a sequence of directed open coverings
of L, then (CCn )n∈N is a sequence of directed covers of L. �

It is known that regular-closed subspaces of almost Menger spaces need not
be almost Menger [20, Example 3.1], but clopen subspaces inherit the almost
Menger property [13, Proposition 3.3]. In frames we present a formally stronger
result. We are going to impose conditions on an onto frame homomorphism
which we first show by an example not to be so stringent as to make the
homomorphism an isomorphism. Recall that an element a of a frame L is co-
linear in case a ∨

∧
i∈Ixi =

∧
i∈I(a ∨ xi) for all families {xi}i∈I of elements of

L.

Example 3.8. Let a be a co-linear element of L and let κa : L → ↑a be the
map given by κa(x) = a ∨ x. Then κa is an onto, weakly perfect (actually,
perfect) frame homomorphism preserving meets (since a is co-linear). Note
though that κa is not an isomorphism if a 6= 0.

Now let us recall from [10, Remark 7.1] that:

if h: L → M is a perfect frame homomorphism, then h∗(a
∗) ≤

h∗(a)
∗ for every a ∈ M.

Proposition 3.9. Let h: L → M be a meet-preserving perfect onto frame
homomorphism. If L is an almost Menger frame, then so is M.

Proof. Let (Cn) be a sequence of directed covers of M. Then (h∗[Cn]) is a
sequence of directed covers of L. Since L is almost Menger, there exists, for
each n, an element un ∈ h∗[Cn] such that the set {un | n ∈ N} has the property
stated in Corollary 3.7. Each un is of the form h∗(cn) for some cn ∈ Cn. We
show that the set {cn | n ∈ N} has the desired property as per Corollary 3.7.
Let a ∈ M. For each n, we can select tn ∈ L such that tn ≥ h∗(cn)

∗ and
h∗(a) =

∧
n∈Ntn. By the result cited above from [10], for each n we have

h(tn) ≥ h
(
h∗(cn)

∗
)
≥ h

(
h∗(c

∗
n)
)
= c∗n

because h is onto. Now, using the fact that h preserves meets, we see that the
elements h(tn) of M, for n ∈ N have the property that

a = h(h∗(a)) =
∧

n∈N

h(tn) and h(tn) ≥ c
∗
n for each n ,

so it follows that M is almost Menger. �

Recall that a coframe is a complete lattice in which binary joins distributive
over meets. A frame which is simultaneously a coframe need not be Boolean.

Corollary 3.10. If L is almost Menger, then cL(a) is almost Menger for every
co-linear a ∈ L. In particular, every closed sublocale of an almost Menger frame
which is also a coframe is almost Menger.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 215



T. Bayih, T. Dube and O. Ighedo

We saw in the proof of Proposition 2.8 that if A is a sublocale of L, then
A is Menger if and only if whenever (Un) is a sequence of families of open
sublocales of L such that A ⊆

∨
Un for every n, then there exists, for each n, a

finite Vn ⊆ Un such that A ⊆
∨⋃

n∈NVn. We present an almost similar result
for the almost Menger property, but only for dense complemented sublocales.
In the proof we shall use the fact that if S is a dense sublocale of L and U
is an open sublocale of L, then S ∩ U = U [15, XIII.1.2.3]. Let us also recall
that if T ⊆ S are sublocales of L, then the closure of T in S is given by
clS T = S ∩ T [15, III.8.5]. We also recall that if S is a sublocale of L and
(Ti)i∈I is a family of sublocales of S, then

S(S)∨
{Ti | i ∈ I} =

S(L)∨
{Ti | i ∈ I}.

In the upcoming proof, the unadorned joins will be in S(L).

Theorem 3.11. The following are equivalent for a complemented dense sublo-

cale A of L.

(1) A is almost Menger.
(2) Whenever (Un) is a sequence of families of open sublocales of L with

A ⊆
∨

Un for every n, then there exists, for each n, a finite Vn ⊆ Un
such that A ⊆

∨{
V | V ∈

⋃
n∈NVn

}
.

Proof. Assume that A is almost Menger, and let (Un) be a sequence of families
of open sublocales of L with A ⊆

∨
Un for every n. Using the notation in the

proof of Proposition 2.8, we have that (A∩ Un) is a sequence of open coverings
of A. Since A is almost Menger, for each n, there is a finite Vn ⊆ Un such that

A =

S(A)∨ {
clA(A ∩ V ) | V ∈

⋃

n∈N

Vn

}

=

S(A)∨ {
A ∩ A ∩ V | V ∈

⋃

n∈N

Vn

}

⊆
∨{

V | V ∈
⋃

n∈N

Vn

}
,

which shows that (1) implies (2).
Conversely, assume that (2) holds. Let (Cn) be a sequence of open coverings

of A. Then, for each n, there is a family Un of open sublocales of L such that
Cn = A ∩ Un. Then A ⊆

∨
Un. By hypothesis, there exists, for each n, a finite

Vn ⊆ Un such that A ⊆
∨{

V | V ∈
⋃

n∈NVn

}
, so that

(‡) A =
∨{

A ∩ V | V ∈
⋃

n∈N

Vn

}

because A is complemented. For each n, put

Dn = {A ∩ V | V ∈ Vn},

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 216



On the Menger and almost Menger properties in locales

and observe that Dn is a finite subset of Cn. Now, for each V ∈
⋃

n∈NVn,

A ∩ V = A ∩ A ∩ V because A ∩ V = V as A is dense and V is open. Since
{
A ∩ V | V ∈

⋃

n∈N

Vn

}
=
{
D | D ∈

⋃

n∈N

Dn

}
,

we deduce from (‡) that

A =

S(A)∨ {
A ∩ D | D ∈

⋃

n∈N

Dn

}
=

S(A)∨ {
clA D | D ∈

⋃

n∈N

Dn

}
,

which shows that A is almost Menger. �

As in spaces, regular almost Menger frames are Menger. In fact, there is a
stronger result. Note that if D is a strong refinement of a cover C of a frame
L, then the open covering D = {o(d) | d ∈ D} of L has the property that for
each U ∈ D there exists a V ∈ C , where C = {o(c) | c ∈ C}, such that U ⊆ V

because d ≺ c implies o(d) ⊆ o(c). As in the case of covers, let us say an open
covering U is a strong refinement of an open covering V if for each U ∈ U
there is a V ∈ V such that U ⊆ V .

Proposition 3.12. A cover regular almost Menger frame is Menger.

Proof. Let L be a cover regular almost Menger frame, and let (Cn) be a se-

quence of open coverings of L. For each n, let Ĉn be a strong refinement of Cn,

so that we have the sequence (Ĉn) of open coverings of L. Since L is almost

Menger, we can choose, for each n, a finite Vn ⊆ Ĉn such that
∨{

V | V ∈
⋃

n∈N

Vn

}
= L.

Since each Ĉn is a strong refinement of Cn, and since Vn is finite, there is a
finite Dn ⊆ Cn such that the closure of each sublocale in Vn is contained in
some sublocale in Dn. Consequently,

L =
∨{

V | V ∈
⋃

n∈N

Vn

}
⊆
∨{

D | D ∈
⋃

n∈N

Dn

}
⊆ L,

showing that (Dn) is a Menger witness for (Cn). Therefore L is Menger. �

We shall now present a result which is, in a way, an analogue of [20, Propo-
sition 3.7]. We recall some pertinent terminology and facts. A frame is called
scattered [18] just in case every sublocale of it is complemented. As observed
in [18, p. 315]:

If f : L → M is a localic map with M scattered, then f−1

[∨
i∈ISi

]
=

∨
i∈If−1[Si] for every family {Si}i∈I of sublocales of M.

A localic map f : L → M is called nearly open if f−1[V ] = f−1[V ] for every
open sublocale V of M. This is a conservative extension of Pták’s [19] notion
of nearly open continuous maps, and it is equivalent to saying h(a∗) = h(a)∗

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 217



T. Bayih, T. Dube and O. Ighedo

for every a ∈ L, where h is the left adjoint of f. Incidentally, this latter
condition was used by Banaschewski and Pultr [3] to define nearly open frame
homomorphisms. Recall that a localic map is called proper if it is closed and
preserves directed joins.

Theorem 3.13. Let f : L → M be a nearly open proper map of locales. Sup-
pose that M is scattered and covered by its compact sublocales. If M is almost
Menger, then L is almost Menger.

Proof. Let (Cn) be a sequence of directed open coverings of L. Write the set
of the compact sublocales of M as an indexed family {Kα | α ∈ A}. Since f
is a proper map, each f−1[Kα] is a compact sublocale of L [21, Corollary 4.3].
Fix n ∈ N. Since Cn is an open covering of L, for each α ∈ A we have the
containment f−1[Kα] ⊆

∨
Cn, which, by compactness and the fact that Cn is

an increasing open covering of L, implies f−1[Kα] ⊆ Cnα, for some Cnα ∈ Cn.
Since Cnα is an open sublocale of L, there exists an element cnα ∈ L such that
Cnα = oL(cnα). Since M is scattered and covered by its compact sublocales,

L = f−1[M] = f−1

[ ∨

α∈A

Kα

]
=
∨

α∈A

f−1[Kα] ⊆
∨

α∈A

oL(cnα),

which implies that the collection Un = {oL(cnα) | α ∈ A} is an open covering
of L. We show from this that the collection

Wn = {oM(f(cnα)) | α ∈ A}

is an open covering of M; and the idea for that is to show that, for each α ∈ A,
Kα ⊆ oM(f(cnα)). Since f−1[Kα] ⊆ oL(cnα), upon taking supplements, we
have

cL(cnα) = L r oL(cnα) ⊆ L r f−1[Kα].

Taking direct images, and using the fact that f is a closed map, we obtain

cM (f(cnα)) = f[cL(cnα)] ⊆ f
[
L r f−1[Kα]

]
⊆ M r Kα;

where the last containment is obtained from [9, Equation (5.2)]. Since every
sublocale of M is complemented, taking supplements in the containment above
yields

Kα = M r (M r Kα) ⊆ M r cM (f(cnα)) = oM(f(cnα)),

whence we deduce that Wn is an open covering of M. Since M is almost
Menger, the sequence (Wn) has an almost Menger witness, (W

′
n), say. Thus,

for each n ∈ N, there exists some k(n) ∈ N and indices α(n,1), . . . , α(n,k(n)) in
A such that

W
′
n =

{
oM
(
f(cn,α(n,1))

)
, . . . , oM

(
f(cn,α(n,k(n)))

)}

and

(#)
∨{

W | W ∈
⋃

n∈N

W
′
n

}
= M.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 218



On the Menger and almost Menger properties in locales

We claim that the sequence (C ′n), where, for each n,

C
′
n =

{
oL(cn,α(n,1)), . . . , oL(cn,α(n,k(n)))

}

is an almost Menger witness for (Cn). It is clear that each C
′
n is a finite subset

of Cn. Since M is scattered and f is nearly open, from (#) we obtain

L = f−1

[∨{
W | W ∈

⋃

n∈N

W
′
n

}]
=
∨{

f−1
[
W
]
| W ∈

⋃

n∈N

W
′
n

}
.

Now, if W ∈ W ′m for some m ∈ N, then W = oM
(
f(cm,α(m,i))

)
, for some

i ∈ {1, . . . , k(m)}, so that W = cM
(
f(cm,α(m,i))

∗
)
. Since h(f(b)) ≤ b, so that

b∗ ≤ h(f(b))∗, for any b ∈ M, and since f is nearly open, we therefore have

f−1
[
W
]
= cL

(
h(f(cm,α(m,i))

∗)
)
= cL

(
h(f(cm,α(n,i)))

∗
)

⊆ cL(c
∗

m,α(m,i))

⊆
∨{

D | D ∈
⋃

n∈N

C
′
n

}
,

because oL(cm,α(m,i)) ∈ C
′
m. It follows therefore that
∨{

D | D ∈
⋃

n∈N

C
′
n

}
= L,

which proves that L is almost Menger. �

The condition that M is scattered was used, among other things, to ensure
that joins (actually, only those that cover the codomain) are preserved under
pullback. It is well known that coverings are generally not preserved under
pullback. However, since in the first part of the proof it is a special type of a
covering (by compact sublocales) that is pulled back along a special type of a
localic map (a nearly open one), it is perhaps worth pointing out that we have
not over hypothesised by requiring the codomain to be scattered. Here is an
example demonstrating the point.

Example 3.14. Consider the frame Ω(R), and let j : B(Ω(R)) → Ω(R) be
the inclusion of its smallest dense sublocale. For any frame L, the frame ho-
momorphism (−)∗∗ : L → BL is nearly open because the pseudocomplement
of an element in any dense sublocale calculated in the sublocale is exactly its
pseudocomplement calculated in the frame. Thus j is a nearly open localic
map. For any x ∈ R, denote by x̃ the prime element R r {x} of Ω(R). By
spatiality,

Ω(R) =
∨{

{r̃, 1Ω(R)} | r ∈ R
}
,

so that Ω(R) is covered by its (compact) one-point sublocales. For any r ∈
R, the set-theoretic inverse image j−1[{r̃, 1Ω(R)}] = {1Ω(R)} because r̃

∗∗ 6= r̃.
Hence j−1[{r̃, 1Ω(R)}] = O, which then says

O =
∨

r∈R

j−1[{r̃, 1Ω(R)}] whereas j−1

[∨

r∈R

{r̃, 1Ω(R)}
]
= B(Ω(R)),

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 219



T. Bayih, T. Dube and O. Ighedo

showing that, generally, j−1 fails to preserve joins of (covering) compact sublo-
cales.

We close with a result that shows that we can replace open coverings with
regular-open coverings in the definition of almost Menger frames. To recall, a
sublocale of L is called regular-open if it is of the form oL(a) with a = a

∗∗. An
element of L of the form x∗∗ is called regular. Clearly, the bijection between
covers and open coverings restricts to a bijection between covers consisting
entirely of regular elements and open coverings consisting entirely of regular-
open sublocales. For any cover C we will write C∗∗ = {c∗∗ | c ∈ C}, and
observe that C∗∗ is also a cover, consisting of regular elements.

Proposition 3.15. A frame is almost Menger iff every sequence of open cov-

erings consisting entirely of regular-open sublocales has an almost Menger wit-

ness.

Proof. Only one implication needs proving. We do it via covers. So, suppose
that every sequence of covers of L consisting entirely of regular elements has
an almost Menger witness. Let (Cn) be a sequence of covers of L, and then
consider the sequence (C∗∗n ). By our supposition, (C

∗∗
n ) has an almost Menger

witness; so, for each n, there exists a finite Un ⊆ C
∗∗
n such that any a ∈ L is

expressible as

a =
∧

α

tα, where each tα ≥ u
∗
α for some uα ∈

⋃

n∈N

Un.

For each n, there exists a positive integer kn and finitely many elements
cn1, . . . , cnkn in Cn such that Un = {c

∗∗
n1, . . . , c

∗∗
nkn

}. Let Dn = {cn1, . . . , cnkn}.
Since (x∗∗)∗ = x∗ always, it follows that (Dn) is an almost Menger witness for
(Cn). Therefore L is almost Menger. �

Acknowledgements. The second-named author acknowledges funding from

the National Research Foundation of South Africa under Grant 113829.

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