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@ Appl. Gen. Topol. 17, no. 1(2016), 51-55doi:10.4995/agt.2016.4376
c© AGT, UPV, 2016

On topological groups with remainder

of character k

Maddalena Bonanzinga and Maria Vittoria Cuzzupé

Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Uni-

versitá di Messina, Italia (mbonanzinga@unime.it, mcuzzupe@unime.it)

Abstract

In [A.V. Arhangel’skii and J. van Mill, On topological groups with
a first-countable remainder, Topology Proc. 42 (2013), 157-163] it is
proved that the character of a non-locally compact topological group
with a first countable remainder doesn’t exceed ω1 and a non-locally
compact topological group of character ω1 having a compactification
whose remainder is first countable is given. We generalize these results
in the general case of an arbitrary infinite cardinal κ.

2010 MSC: 54H11; 54A25; 54B05.

Keywords: character; compactification; π-base; remainder; topological
group.

1. Introduction

In [3] the authors answer in the negative to the following problem:

Problem 1.1 ([3, Problem 1.1]). Suppose that G is a non-locally compact topo-
logical group with a first countable remainder. Is G metrizable?

Also, the following necessary condition for a non-locally compact topological
group to have a first countable remainder is established:

Theorem 1.2 ([3, Theorem 2.1]). Suppose that G is a non-locally compact
topological group with a first countable remainder. Then the character of the
space G doesn’t exceed ω1.

As a consequence of the previous result the following holds:

Received 23 November 2015 – Accepted 04 January 2016

http://dx.doi.org/10.4995/agt.2016.4376


M. Bonanzinga and M. V. Cuzzupé

Theorem 1.3 ([3, Theorem 2.4]). If G is a non-locally compact topological
group with a first countable remainder, then |G| ≤ 2ω1.

In [3] it is proved that Theorem 1.2 is the best possible giving the following:

Example 1.4 ([3, Section 3]). A non locally compact topological group G of
character ω1 which has a compactification bG such that bG\G is first countable.

In this paper we show that the methods used by Arhangel’skii and van Mill
permit to generalize Theorem 1.2 and Example 1.4 in the general case of an
arbitrary infinite cardinal κ.

By a space we understand a Tychonoff topological space. By a remainder of
a space we mean the subspace bX \ X of a Hausdorff compactification bX of
X. We follow the terminology and notation in [4].

2. Generalizations of Arhangel’skii and van Mill’s results in the
general case of an arbitrary infinite cardinal κ

We show that Arhangel’skii and van Mill’s proof of [3, Theorem 2.1], works
in the general case of an arbitrary infinite cardinal κ.

Theorem 2.1. Let κ be an infinite cardinal and let G be a non-locally com-
pact topological group. Assume that G has a compactification such that its
remainder bG \ G has character κ. Then the character of the space G doesn’t
exceed κ+.

To prove Theorem 2.1, we need the following propositions 2.2 and 2.3. In
particular, Proposition 2.2 is known (see for example [1], also note that the
concept of free sequence was introduced in [2]). We include the proof of Propo-
sition 2.2 for completness of the exposition.

Proposition 2.2. Suppose that Y is a space with tightness t(Y ) = κ satisfying
the following condition:

(s) for any subset A of Y such that |A| ≤ κ+, the closure of A in Y is
compact.

Then Y is compact.

Proof. Striving for a contradiction, assume that Y is not compact and let X
be a compactification of Y . Pick an arbitrary point x ∈ X \Y . Then:

Fact 1: Every nonempty Gκ-subset P of X that contains x meets Y .

Indeed, let P =
⋂
{Vα : α < κ}, where each Vα is open. For each α take an

open set Uα in X such that x ∈ Uα ⊆ Vα. Put {Uα : α < κ} = U. We may
assume without any loss of generality that U is closed under finite intersections.
For any U ∈U pick a point yU ∈ U∩Y and let A = {yU : U ∈U}. By condition

(s), the set S = A
Y

is compact. As the family F = {U ∩ S : U ∈ U} has the
finite intersection property, we must have

⋂
F 6= ∅. Since

⋂
F ⊆ P ∩ Y , we

are done.
Using Fact 1, we define for every ξ < κ+ a point yξ ∈ Y and a closed Gκ-

subset Pξ of X containing x, as follows. Let y0 be any element of Y , and put

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On topological groups with remainder of character k

P0 = X. Now assume that ξ < κ
+, and that the points yβ ∈ Y and the closed

Gκ-subsets Pβ of X have been defined for every β < ξ. Denote by Fξ the
closure of the set {yβ : β < ξ} in X. Then Fξ ⊆ Y and x /∈ Fξ. Since Fξ is
closed in X and X is Tychonoff, it follows that there exists a closed Gδ-subset
V of x in X such that x ∈ V and V ∩Fξ = ∅. Put Pξ = V ∩

⋂
β<ξ

Pβ. Clearly,
x ∈ Pξ, and Pξ is a closed Gκ-subset of X. By Fact 1, we have Pξ ∩ Y 6= ∅.
This completes the transfinite construction.

Obviously, the following statements hold for any ξ < κ+ (Fact 4 follows
directly from facts 2 and 3).

Fact 2: {yβ : β < ξ}∩ Pξ = ∅.

Fact 3: {yβ : ξ ≤ β < κ+}⊆ Pξ.

Fact 4: {yβ : β < ξ}∩{yβ : ξ ≤ β < κ+} = ∅.

Fact 4 implies that η = {yξ : ξ < κ
+} is a free sequence in X. Its closure

is compact and is contained in Y . Hence this contradicts the fact that the
tightness of Y is at most κ (Juhász [5, 3.12]). �

Following the argument from [3, Proposition 2.3], and using Proposition 2.2
instead of [3, Proposition 2.2] we obtain the following result.

Proposition 2.3. Suppose that X is a nowhere locally compact space with
remainder Y such that χ(Y ) = κ, where κ is an infinite cardinal. Then the
π-character of the space X doesn’t exceed κ+ at some point of X.

Proof of Theorem 2.1. It follows from Proposition 2.3 that there exists a π-
base P of G at the neutral element e of G such that |P| ≤ κ+. Then, clearly,
the family µ = {UU−1 : U ∈P} is a base of G at e such that |µ| ≤ κ+. �

Theorem 2.4. If G is a non-locally compact topological group with remainder

Y such that χ(Y ) = κ, then |G| ≤ 2κ
+

.

Proof. Let bG be a compactification of the space G such that the remainder
Y = bG \ G has character κ. By Theorem 2.1, the character of the space G
doesn’t exceed κ+. Since χ(Y ) = κ and Y and G are both dense in bG, we

conclude that χ(bG) ≤ κ+. Since bG is compact, it follows that |bG| ≤ 2κ
+

.

Hence, |G| ≤ 2κ
+

. �

Following the method in [3, Section 3] we construct the following example.

Example 2.5. A non-locally compact topological group G of character κ+

which has a compactification bG such that bG\G has character κ.

Let X be a space with a dense subset D and consider the subspace

X(D) = (X ×{0})∪ (D ×{1})

of the Alexandroff duplicate of X.
Observe that X(D) is compact if X is compact.

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M. Bonanzinga and M. V. Cuzzupé

The idea used by authors in [3, Section 3] is to replace every isolated point of
the form (d, 1) in X(D) by a copy of a fixed non-empty space Y . Also they note
that if both X and Y are compact, then so is X(D, Y ) and that the function
π : X(D, Y ) → X ×{0} that collapses each set of the form {d}× Y ×{1} to
(d, 0) is a retraction.

Let κ ≥ ω and let K = 2κ(2κ), i.e., the Alexandroff duplicate of the Cantor
cube 2κ. Following the idea used in [3, Section 3] and using this building block
repeatedly, we will construct an inverse sequence of compact spaces Xα, α <
κ+.

In particular following step by step [3, Section 3] and defining X0 = 2
κ

instead of 2ω, we construct all Xα, where α < ω1 and Xω1 = lim←−
{Xα, π

α
β}. Let

πω1α : Xω1 → Xα denotes the projection for all α < ω1.
Also the points p ∈ Xω1, for which π

ω1
α (p) is isolated for every successor

ordinal number α < ω1, form a dense subspace H in Xω1.
Now put Xω1+1 = Xω1(H), and let π

ω1+1
ω1

be the standard retraction. We
continue as before, replacing each isolated point by a copy of K, etc. Let
Xω1+ω1 be the inverse limit of spaces Xω1+β, β < ω1. Continuing in this way
for all α < ω2, we get an inverse sequence {Xα, π

α
β} of compact spaces having

character equal to κ. Let Xω2 = lim←−
{Xα, π

α
β} with retractions π

ω2
α : Xω2 → Xα

for all α < ω2.
Continuing in this way for all α < κ+, we get an inverse sequence {Xα, π

α
β}

of compact spaces having character equal to κ. Let Xκ+ = lim←−
{Xα, π

α
β} with

retractions πκ
+

α : Xκ+ → Xα for all α < κ
+.

The following fact holds.

If p ∈ Xκ+ and there exists a successor ordinal number α < κ
+ such that

πκ
+

α (p) is not isolated, then

(πκ
+

α )
−1({πκ

+

α (p)}) = {p}.

Hence, Xκ+ has character equal to κ at p.

The points p ∈ Xκ+, for which π
κ
+

α (p) is isolated for every successor ordinal
number α < κ+, form a dense subspace G in Xκ+. The space G is easily

seen to be homeomorphic to the space (2κ)κ
+

with the Gκ-topology (where the
topology on 2κ is the standard product topology). The reason that we get the
Gκ-topology is clear: because if p ∈ G, then, for every α < κ

+, we have that

πκ
+

α+1(p) is isolated. Hence, G is a topological group, and so we are done.

3. Suggestions for further research

It seems natural to pose the following question:

Question 3.1. Is it possible to generalize Theorem 2.1 and Example 2.5 in
some “generalized” class of topological groups? In particular, is this possible
in the class of paratopological groups?

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On topological groups with remainder of character k

Acknowledgements. The authors express gratitude to Jan van Mill for use-

ful communications.

References

[1] A. V. Arhangel’skii, Construction and classification of topological spaces and cardinal
invariants, Uspehi Mat. Nauk. 33, no. 6 (1978), 29–84.

[2] A. V. Arhangel’skii, On the cardinality of bicompacta satisfying the first axiom of count-
ability, Doklady Acad. Nauk SSSR 187 (1969), 967–970.

[3] A. V. Arhangel’skii and J. van Mill, On topological groups with a first-countable re-
mainder, Topology Proc. 42 (2013), 157–163.

[4] R. Engelking, General Topology, Heldermann Verlag, Berlin, second ed., 1989.
[5] I. Juhász, Cardinal functions in topology–ten years later, Mathematical Centre Tract,

vol. 123, Mathematisch Centrum, Amsterdam, 1980.

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