Extension of mappings from the product of pseudocompact spaces

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Extension of mappings from the product of

pseudocompact spaces

Evgenii Reznichenko

Department of General Topology and Geometry, Mechanics and Mathematics Faculty,

M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 199991 Russia

Abstract

Let 𝑋and 𝑌be pseudocompact spaces and let a function Φ : 𝑋×𝑌→R

be separately continuous. The following conditions are equivalent: (1) there

is a dense 𝐺𝛿subset of 𝐷⊂𝑌such that Φis continuous at every point of

𝑋×𝐷(Namioka property); (2) Φis quasicontinuous; (3) Φextends to a sep-

arately continuous function on 𝛽 𝑋 ×𝛽 𝑌 . This theorem makes it possible to

combine studies of the Namioka property and generalizations of the Eberlein-

Grothendieck theorem on the precompactness of subsets of function spaces. We

also obtain a characterization of separately continuous functions on the product

of several pseudocompact spaces extending to separately continuous functions

on products of Stone–ˇ

Cech extensions of spaces. These results are used to study

groups and Mal’tsev spaces with separately continuous operations.

Keywords: Extension of functions, Stone-ˇ

Cech extension, Pseudocompact

spaces, Quasi-continuous functions, Mal’tsev spaces, Eberlein-Grothendieck

theorem

1. Introduction

Let 𝑋and 𝑌be topological spaces. A function Φ : 𝑋×𝑌→Ris separately

continuous if the functions Φ(·, 𝑦) : 𝑋→Rand Φ(𝑥, ·) : 𝑌→Rare continuous

for 𝑥∈𝑋and 𝑦∈𝑌. The consideration of separate continuity vis-`a-vis joint

continuity goes back, at least, to Bare 1899 [1], whose work is the prototype of

all the subsequent investigations on this subject by many mathematicians.

A function 𝑓:𝑍→Ris called quasi-continuous if for every point 𝑧∈𝑍,

neighborhood 𝑂of 𝑓(𝑧), neighborhood 𝑊of 𝑧there exists a non-empty open

𝑈⊂𝑊such that 𝑓(𝑈)⊂𝑂.

The following continuity conditions for the function Φare considered.

(𝐶1) There is a dense type 𝐺𝛿subset 𝐷⊂𝑌=𝐷such that Φis (jointly)

continuous at every point (𝑥, 𝑦)∈𝑋×𝐷[2].

Email address: erezn@inbox.ru (Evgenii Reznichenko)

Preprint submitted to Elsevier November 8, 2022

arXiv:2210.10923v3 [math.GN] 7 Nov 2022

(𝐶2) The function Φis quasi-continuous.

(𝐶3) The function Φextends to a separately continuous function 

Φ : 𝛽 𝑋 ×𝑌→

R, where 𝛽 𝑋 is the Stone–ˇ

Cech extension of 𝑋[3].

Clearly, (𝐶1) implies (𝐶2). Spaces 𝑋and 𝑌satisfy the Namioka property

𝒩(𝑋, 𝑌 )if for every separately continuous map Φthe condition (𝐶1) is sat-

isﬁed [2]. If the condition (𝐶1) is satisﬁed, then the function Φis also said to

satisfy the Namioka property. We say that (𝑋, 𝑌 )is a Grothendieck pair if for

every separately continuous map Φthe condition (𝐶3) is satisﬁed [3].

Note that [3] gave a diﬀerent deﬁnition: (𝑋, 𝑌 )is a Grothendieck pair if

for every continuous map 𝜙:𝑋→𝐶𝑝(𝑌)the closure of 𝜙(𝑋)in 𝐶𝑝(𝑌)is

compact, where 𝐶𝑝(𝑌)is the space of continuous functions on 𝑌in the topology

of pointwise convergence [3, Deﬁnition 1.7]. Assertion 1.2 of [3] implies that

these two deﬁnitions are equivalent.

An important special case of the general situation is when the spaces 𝑋and

𝑌are pseudocompact. This case was mainly considered in [3, 4]. The main

result of this paper is that in this case, if the function Φis separately con-

tinuous, then the conditions (𝐶1), (𝐶2) and (𝐶3) are equivalent (Theorem 1).

This implies that if the spaces 𝑋and 𝑌are pseudocompact, then (𝑋, 𝑌 )is a

Grothendieck pair if and only if 𝑋and 𝑌satisfy the Namioka property (The-

orem 4). This theorem allows using the Namioka property theorems to ﬁnd

Grothendieck pairs and vice versa, the Grothendieck pair theorems to ﬁnd pairs

of spaces with the Namioka property.

A space 𝑌is called a weakly 𝑝𝑐-Grothendieck space if any pseudocompact

subspace of 𝐶𝑝(𝑌)has a compact closure in 𝐶𝑝(𝑌)[5]. In other words, 𝑌is

a weakly 𝑝𝑐-Grothendieck space if and only if (𝑋, 𝑌 )is a Grothendieck pair

for any pseudocompact space 𝑋. Theorem 4 allows one to ﬁnd new classes of

pseudocompact weakly 𝑝𝑐-Grothendieck spaces.

Using Theorem 1 and the results of [6] in Section 5, we obtain a criterion for

a function of several variables on a product of pseudocompact spaces to extend

to a product of Stone–ˇ

Cech extensions (Theorem 4 and Theorem 5). Using

the results of Section 2 in Section 6, we obtain theorems on the continuity of

operations in Mal’tsev groups and spaces.

The terminology follows the books [7, 8]. By spaces we mean Tikhonoﬀ

spaces.

2. Extension of functions from a product of spaces

Proposition 1. Let 𝑋and 𝑌be pseudocompact spaces and Φ : 𝑋×𝑌→Rbe a

separately continuous quasi-continuous function. Then 𝑓extends to a separately

continuous function 

Φ : 𝛽𝑋 ×𝑌→R.

Proof. Denote by 𝐶the set of points in 𝑋×𝑌at which the function Φis

continuous. Denote 𝐶𝑦={𝑥∈𝑋: (𝑥, 𝑦)∈𝐶}for 𝑦∈𝑌.

Lemma 1. The set 𝐶𝑦is dense in 𝑋for all 𝑦∈𝑌.

Proof. Assume the opposite, i.e. 𝑈′=𝑋∖𝐶𝑦′̸=∅for some 𝑦′∈𝑌. Let us put

Ψ(𝑥, 𝑦) = |Φ(𝑥, 𝑦)−Φ(𝑥, 𝑦′)|

for (𝑥, 𝑦)∈𝑋×𝑌. The function Ψis non-negative, separately continuous, quasi-

continuous, and discontinuous at points of the set 𝑈′× {𝑦′}, and Ψ(𝑥, 𝑦′) = 0

for 𝑥∈𝑋. For 𝑂⊂𝑋×𝑌and (𝑥, 𝑦)∈𝑋×𝑌we set

𝜔Ψ(𝑂) = sup{|Φ(𝑥1, 𝑦1)−Φ(𝑥2, 𝑦2)|: (𝑥1, 𝑦1),(𝑥2, 𝑦2)∈𝑂},

𝜔Ψ(𝑥, 𝑦) = inf{𝜔Ψ(𝑂) : 𝑂is a neighborhood of the point (𝑥, 𝑦)}.

Let us put

𝐹𝑛={(𝑥, 𝑦)∈𝑋×𝑌:𝜔Ψ(𝑥, 𝑦)≥1

2𝑛}, 𝐹 ′

𝑛={𝑥∈𝑋: (𝑥, 𝑦′)∈𝐹𝑛}

for 𝑛∈𝜔. The set 𝐹𝑛is closed in 𝑋×𝑌and 𝐹′

𝑛is closed in 𝑋. The set 𝑛∈𝜔𝐹𝑛

is the set of discontinuity points of the function Ψ, so 𝑈′⊂𝑛∈𝜔𝐹′

𝑛. Since 𝑋

is a Baire space, there exists a non-empty open 𝑈⊂𝑈′∩𝐹′

𝑛for some 𝑛∈𝜔.

We set 𝜀=1

3·2𝑛and

𝑀={(𝑥, 𝑦)∈𝑋×𝑌: Ψ(𝑥, 𝑦)>2𝜀}.

Then 𝑈× {𝑦′} ⊂ 𝑀. Let 𝑈−1=𝑈and 𝑉−1=𝑌. By induction on 𝑛we

construct a sequence

(𝑥𝑛, 𝑉𝑛, 𝑈𝑛, 𝑊𝑛)𝑛∈𝜔,

where 𝑥𝑛∈𝑈,𝑉𝑛⊂𝑌is an open neighborhood of 𝑦′,𝑈𝑛⊂𝑈is an open

non-empty set, 𝑊𝑛⊂𝑌is an open non-empty set such that for every 𝑛∈𝜔the

following conditions are met:

(1) 𝑥𝑛∈𝑈𝑛and 𝑈𝑛⊂𝑈𝑛−1;

(2) 𝑦′∈𝑉𝑛,𝑉𝑛⊂𝑉𝑛−1and 𝑊𝑛⊂𝑉𝑛;

(3) Ψ({𝑥𝑛} × 𝑉𝑛)⊂[0, 𝜀);

(4) Ψ(𝑈𝑛×𝑊𝑛)⊂(2𝜀, +∞).

Let us carry out the construction at the 𝑛th step. Since 𝑈×{𝑦′} ⊂ 𝑀,𝑦′∈𝑉𝑛−1

and 𝑈𝑛−1⊂𝑈, there exists (𝑥′′, 𝑦′′)∈𝑀∩(𝑈𝑛−1×𝑉𝑛−1). Then Ψ(𝑥′′, 𝑦′′)>2𝜀.

Since the function is quasi-continuous, Ψ(𝑈𝑛×𝑊𝑛)⊂(2𝜀, +∞)for some non-

empty open 𝑈𝑛⊂𝑈𝑛⊂𝑈𝑛−1and 𝑊𝑛⊂𝑊𝑛⊂𝑉𝑛−1. Take 𝑥𝑛∈𝑈𝑛. We

choose a neighborhood 𝑉𝑛of the point 𝑦′in such a way that 𝑉𝑛⊂𝑉𝑛−1and

Ψ({𝑥𝑛} × 𝑉𝑛)⊂[0, 𝜀).

Let 𝐺=𝑛∈𝜔𝑈𝑛. Since 𝑋is pseudocompact, then 𝐺is a non-empty closed

subset of 𝑋. Since 𝑌is pseudocompact, the sequence (𝑊𝑛)𝑛∈𝜔accumulates to

some point 𝑦*∈𝑌. We put 𝑓(𝑥) = Ψ(𝑥, 𝑦*)for 𝑥∈𝑋. The function 𝑓:𝑋→R

is continuous. It follows from (2) that 𝑦*∈𝑄=𝑛∈𝜔𝑉𝑛. It follows from (4)

that 𝑓(𝐺)⊂[2𝜀, +∞). Since 𝑦*∈𝑉𝑛, it follows from (3) that 𝑓(𝑥𝑛)< 𝜀

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摘要：

ExtensionofmappingsfromtheproductofpseudocompactspacesEvgeniiReznichenkoDepartmentofGeneralTopologyandGeometry,MechanicsandMathematicsFaculty,M.V.LomonosovMoscowStateUniversity,LeninskieGory1,Moscow,199991RussiaAbstractLet�and�bepseudocompactspacesandletafunction:�...

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