@
Appl. Gen. Topol. 23, no. 2 (2022), 303-314

doi:10.4995/agt.2022.16925

© AGT, UPV, 2022

Cardinal invariants and special maps of

quasicontinuous functions with the topology of

pointwise convergence

Mandeep Kumar a and Brij Kishore Tyagi b

a Department of Mathematics, University of Delhi, Delhi - 110007, India. (mjakhar5@gmail.com)
b Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi, New

Delhi - 110021, India. (brijkishore.tyagi@gmail.com)

Communicated by D. N. Georgiou

Abstract

For topological spaces X and Y , let Qp(X, Y ) be the space of all qua-
sicontinuous functions from X to Y with the topology of pointwise
convergence. In this paper, we study the cardinal invariants such as
character, weight, density, pseudocharacter, spread and cellularity of
the space Qp(X, Y ). We also discuss the properties of the restriction
and induced maps related to the space Qp(X, Y ).

2020 MSC: 54C35; 54C08; 54C30.

Keywords: quasicontinuous functions; topology of pointwise convergence;
character; weight; density; spread; cellularity; induced map; re-
striction map.

1. Introduction

Kempisty [10] introduced a weaker form of continuity for real-valued func-
tions, named as quasicontinuity. The properties of quasicontinuous functions
are discussed in many papers, for example see [2, 13, 15, 16].

The quasicontinuous functions have various applications in different areas of
mathematics; for instance topological groups [11], dynamical systems [4] and
the study of minimal usco and minimal cusco maps [6]. Some examples [3] of

Received 23 December 2021 – Accepted 3 April 2022

http://dx.doi.org/10.4995/agt.2022.16925
https://orcid.org/0000-0001-8773-0927
https://orcid.org/0000-0003-2660-2432


M. Kumar and B. K. Tyagi

quasicontinuous functions are the doubling function
D : [0, 1) → [0, 1) defined by D(x) = 2x (mod 1),

the extended sin(1/x) function f : R → R defined by

f(x) =

{
sin( 1

x
) if x 6= 0

0 if x = 0,

the floor function from R to R defined by
bxc = max{n ∈ Z : n ≤ x},

and any monotonic left or right continuous function from R to R [15].
The set of all real-valued quasicontinuous maps on a topological space X with

the topology of pointwise convergence, denoted by Qp(X,R), is studied in [7, 8,
9]. The pointwise convergence of real-valued quasicontinuous maps defined on a
Baire space is examined in [7]. In [9] metrizability, first countability, closed and
compacts subsets of the space Qp(X,R) are discussed. The cardinal functions
of the space Qp(X,R) are studied in [8].

In this paper, we study the results about metrizability, first countability and
cardinal functions of the space Qp(X,Y ) along with the concept of induced and
the restriction maps.

In a more detail, this paper is organized as follows: In Section 3, we define the
topology of pointwise convergence on Q(X,Y ), the set of all quasicontinuous
functions from a topological space X to a topological space Y . In Section 4,
when X is Hausdorff and Y is a nontrivial T1-space, we compare the cardinal
functions π-character, character and weight of the space Qp(X,Y ). Moreover,
if Y is second countable, we characterize these cardinal functions. For a regular
space X and a nontrivial T1-space Y , we discuss pseudocharacter and spread
of the space Qp(X,Y ). We also show that Qp(X,Y ) is dense in the space Y

X.
In Section 5, we discuss the topological properties of induced maps and the
restriction map related to the space Qp(X,Y ).

2. Preliminaries

Throughout this paper, the symbols X,Y,Z are topological spaces unless
otherwise stated, R is the space of real numbers with the usual topology, N is
the set of positive integers, and I is the closed interval [−1, 1]. The topology
of a space X is denoted by τ(X). By a nontrivial space we mean a topological
space with at least two different points. The symbol Ao denotes the interior of
A in X and the symbol A denotes the closure of A in X.

Definition 2.1. A map f : X → Y is quasicontinuous [15] at x ∈ X if for every
open set U containing x and every open set V containing f(x), there exists a
nonempty open set G ⊆ U such that f(G) ⊆ V . If f is quasicontinuous at
every point of X, we say that f is quasicontinuous.

Note that every continuous map is quasicontinuous. Conversely, for X =
[0, 1) with the usual topology and Y = [0, 1) with the Sorgenfrey topology, the
identity map from X to Y is quasicontinuous but nowhere continuous [12].

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 304



Cardinal invariants and special maps of quasicontinuous functions

Levine [13] studied quasicontinuous maps under the name of semi-continuity
using the terminology of semi-open sets. A subset A of a space X is said to be
semi-open (or quasi-open [15]) if A ⊆ Ao. A map f : X → Y is quasicontinuous
if and only if for every open set V in Y , f−1(V ) is semi-open in X.

3. Quasicontinuous functions and the topology of pointwise
convergence

Let F(X,Y ) be the set of all functions and C(X,Y ) be the set of all contin-
uous functions from X to Y . The function spaces F(X,Y ) and C(X,Y ) with
the topology of pointwise convergence denoted by Fp(X,Y ) and Cp(X,Y ), re-
spectively, are widely studied in the literature, for example [1, 5, 14, 17].

For x ∈ X and V ∈ τ(Y ), let S(x,V ) = {f ∈ F(X,Y ) : f(x) ∈ V}. Then
Fp(X,Y ) has a subbase

S = {S(x,V ) : x ∈ X,V ∈ τ(Y )}.
Note that F(X,Y ) = Y X and the topology of pointwise convergence on F(X,Y )
is just the product topology on Y X.

Let Q(X,Y ) be the set of all quasicontinuous functions in F(X,Y ). The
space Q(X,Y ) with the topology of pointwise convergence is the subspace
Q(X,Y ) of the space Fp(X,Y ) and is denoted by Qp(X,Y ).

For x ∈ X and V ∈ τ(Y ), denote [x,V ] = {f ∈ Q(X,Y ) : f(x) ∈ V}. Then
S′ = {[x,V ] : x ∈ X,V ∈ τ(Y )} is a subbase for the space Qp(X,Y ). Observe
that for x ∈ X and V1,V2 ∈ τ(Y ), we have [x,V1] ∩ [x,V2] = [x,V1 ∩V2]. For
x1, . . . ,xn ∈ X and V1, . . . ,Vn ∈ τ(Y ), denote

[x1, . . . ,xn; V1, . . . ,Vn] = {f ∈ Q(X,Y ) : f(xi) ∈ Vi, 1 ≤ i ≤ n}.
Clearly the family

B = {[x1, . . . ,xn; V1, . . . ,Vn] : xi ∈ X,Vi ∈ τ(Y ), 1 ≤ i ≤ n,n ∈ N}
is a base for the space Qp(X,Y ). If V is a basis for Y then the family

B′ = {[x1, . . . ,xn; V1, . . . ,Vn] : xi ∈ X,Vi ∈V, 1 ≤ i ≤ n,n ∈ N}
is also a basis for the space Qp(X,Y ). If (Y,d) is a metric space then for
f ∈ Q(X,Y ); x1, . . . ,xn ∈ X and � > 0, denote

O(f,x1, . . . ,xn,�) = {g ∈ Q(X,Y ) : d(g(xi),f(xi)) < �,1 ≤ i ≤ n}.
It is easy to see that the family

Bf = {O(f,x1, . . . ,xn,�) : x1 . . . ,xn ∈ X,n ∈ N,� > 0}
is a local base at f ∈ Qp(X,Y ).

4. Cardinal functions and the space Qp(X, Y )

In this section, we discuss first countability, metrizability and cardinal func-
tions of the space Qp(X,Y ). Before generalizing some results obtained in [8, 9],
first we recall definitions of the cardinal functions for a topological space [8, 14].

A collection V of nonempty open subsets of X is called a local π-base at
x ∈ X if for each open set U containing x, there exists V ∈V such that V ⊆ U.
The π-character of a point x ∈ X is πχ(x,X) = ℵ0 + min{|V| : V is a local π-
base at x}. The π-character of a space X is defined as πχ(X) = sup{πχ(x,X) :

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 305



M. Kumar and B. K. Tyagi

x ∈ X}. The character of a space X is χ(X) = sup{χ(x,X) : x ∈ X}, where
χ(x,X) = ℵ0 + min{|Bx| : Bx is a base at x}. The weight of a space X is
defined by ω(X) = ℵ0 + min{|B| : B is a base for X}. A collection β of
nonempty subsets of a space X is called a π-base for X provided that every
nonempty open subset of X contains some member of β. The π-weight of a
space X is defined by πω(X) = ℵ0 + min{|β| : β is a π-base for X}. The
density of a space X is d(X) = ℵ0 + min{|D| : D is a dense subset of X}. The
pseudocharacter of a point x ∈ X is ψ(x,X) = ℵ0 + min{|γ| : γ is a family
of open sets in X such that ∩γ = {x}}. The pseudocharacter of a space X is
defined as ψ(X) = sup{ψ(x,X) : x ∈ X}. The spread of a space X is defined as
s(X) = ℵ0 + sup{|D| : D ⊆ X is discrete}. The cellularity or Souslin number of
a space X is defined by c(X) = ℵ0 + sup{|U| : U is a family of pairwise disjoint
nonempty open subsets of X}. A space X is said to have Souslin property if
c(X) = ℵ0.

Lemma 4.1 ([8, Lemma 4.2]). Let X and Y be topological spaces and f : X →
Y be a map such that for any x ∈ X, there exists an open set G in X such that
x ∈ G and f(y) = f(x) for all y ∈ G. Then f is quasicontinuous.

Lemma 4.2 ([8, Lemma 4.3]). Let X and Y be topological spaces such that X is
Hausdorff. For given x1, . . . ,xn ∈ X and (not necessarily distinct) y1, . . . ,yn ∈
Y , there exists a quasicontinuous map f : X → Y such that f(xi) = yi for each
i ∈{1, . . . ,n}.

Theorem 4.3. Let X and Y be topological spaces such that X is uncountable
Hausdorff and Y is a nontrivial T1-space. Then for any f ∈ Qp(X,Y ), f does
not have a countable local π-base.

Proof. Suppose {Un : n ∈ N} be a countable local π-base at some f ∈
Qp(X,Y ). So for each n, there is a basic open set Wn such that Wn ⊆
Un. Then {Wn : n ∈ N} is also a countable local π-base at f. Let Wn =
[xn1 , . . . ,x

n
kn

; V n1 , . . . ,V
n
kn

] for each n ∈ N. The set A = {xij : 1 ≤ j ≤ ki, i ∈ N}
is countable.

Since X is uncountable, choose x ∈ X \ A. Because Y is a nontrivial T1-
space, choose y ∈ Y such that y /∈ V for some open set V containing f(x).
Then W = [x,V ] is an open set containing f. Suppose Wn ⊆ W for some
n ∈ N. By Lemma 4.2, let g : X → Y be a quasicontinuous function such
that g(xni ) ∈ V

n
i for each i ∈ {1, . . . ,kn} and g(x) = y. Then g ∈ Wn \W , a

contradiction. �

Corollary 4.4. Let X and Y be topological spaces such that X is Hausdorff
and Y is a nontrivial T1-space. If the space Qp(X,Y ) has a countable local
π-base at some f ∈ Qp(X,Y ), then X is countable.

Corollary 4.5. Let X be an uncountable Hausdorff space and Y be a nontrivial
T1-space. Then for any f ∈ Qp(X,Y ), f does not have a countable local base.

Using Corollary 4.4, a more general result than [9, Theorem 3.2] is the
following.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 306



Cardinal invariants and special maps of quasicontinuous functions

Theorem 4.6. Let X and Y be spaces such that X is Hausdorff and Y is a
nontrivial metrizable space. Then the following are equivalent:

(a) Fp(X,Y ) is metrizable.
(b) Fp(X,Y ) is first countable.
(c) Qp(X,Y ) is metrizable.
(d) Qp(X,Y ) is first countable.
(e) For any f ∈ Qp(X,Y ), f has a countable local π-base.
(f ) X is countable.

Proof. Clearly (d) implies (e) holds. The assertion (e) implies (f) follows from
Corollary 4.4. Using the facts that Qp(X,Y ) is a subspace of Fp(X,Y ) and a
countable product of metrizable spaces is metrizable, the rest of the implica-
tions can be verified easily. �

The result obtained in Theorem 4.3 can also be deduced from the following
result about the cardinal functions related to the space Qp(X,Y ).

Theorem 4.7. Let X and Y be topological spaces such that X is Hausdorff
and Y is a nontrivial T1-space. Then |X| ≤ πχ(Qp(X,Y )) ≤ χ(Qp(X,Y )) ≤
ω(Qp(X,Y )). Moreover, if X is infinite and Y is second countable, we have
|X| = πχ(Qp(X,Y )) = χ(Qp(X,Y )) = πω(Qp(X,Y )) = ω(Qp(X,Y )).

Proof. To show |X| ≤ πχ(Qp(X,Y )), let y1 ∈ Y and f ∈ Qp(X,Y ) be the con-
stant function such that f(x) = y1 for each x ∈ X. Let {Ut : t ∈ T} be a local
π-base at f with |T | ≤ πχ(Qp(X,Y )). Since each Ut is a nonempty open subset
of Qp(X,Y ), there exists a basic open set Bt = [x

t
1, . . . ,x

t
nt

; V t1 , . . . ,V
t
nt

] ⊆ Ut
for each t ∈ T. Then the collection Bf = {Bt : t ∈ T} is also a local π-base at
f. For each t ∈ T, let At = {xt1, . . . ,xtnt}. We claim that

⋃
t∈T At = X.

Let x ∈ X. Since Y is a nontrivial T1-space, choose y2 ∈ Y such that
y2 /∈ V1 for some open set V1 in Y containing y1. Because [x,V1] is an open
set containing f and Bf is a local π-base at f, there exists t ∈ T such that
Bt = [x

t
1, . . . ,x

t
nt

; V t1 , . . . ,V
t
nt

] ⊆ [x,V1]. We claim that x ∈ At = {xt1, . . . ,xtnt}.
Suppose x /∈ At. Since X is Hausdorff, there exists an open set U such

that x ∈ U and U ∩ At = ∅. Because Bt is nonempty, let st ∈ Bt. Then
st(x

t
i) ∈ V

t
i for each i ∈{1, . . . ,nt}. By Lemma 4.2, there is a quasicontinuous

map ht : X → Y such that ht(xti) = st(x
t
i) for each i ∈ {1, . . . ,nt}. Let us

define g : X → Y such that

g(z) =

{
y2 if z ∈ U
ht(z) if z ∈ X \U

By Lemma 4.1, g is a quasicontinuous map such that g ∈ Bt, but g /∈ [x,V1],
which contradicts Bt ⊆ [x,V1]. So

⋃
t∈T At = X. Hence |X| ≤ πχ(Qp(X,Y )) ≤

χ(Qp(X,Y )) ≤ ω(Qp(X,Y )).
If BY is a countable base for Y then {[x1, . . . ,xn; V1, . . . ,Vn] : xi ∈ X,Vi ∈

BY , 1 ≤ i ≤ n} is a base for the space Qp(X,Y ). Thus ω(Qp(X,Y )) ≤ |X|. �

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 307



M. Kumar and B. K. Tyagi

Theorem 4.8. Let X and Y be topological spaces such that X is infinite Haus-
dorff space and Y is a nontrivial metrizable space. Then |X| = πχ(Qp(X,Y )) =
χ(Qp(X,Y )).

Proof. By Theorem 4.7, we have |X| ≤ πχ(Qp(X,Y )) ≤ χ(Qp(X,Y )). To
show χ(Qp(X,Y )) ≤ |X|, let f ∈ Qp(X,Y ). If (Y,d) is a metric space then the
collection Bf = {O(f,x1, . . . ,xk, 1n) : x1, . . . ,xk ∈ X,k,n ∈ N} is a local base
at f. Thus |X| = πχ(Qp(X,Y )) = χ(Qp(X,Y )). �

Lemma 4.9. Let X and Y be topological spaces such that U1, . . . ,Un are
nonempty pairwise disjoint open subsets of X and y1, . . . ,yn ∈ Y . Then there
exists a quasicontinuous map g : X → Y such that g(Ui) = {yi} for each
i ∈{1, . . . ,n}.

Proof. Let H = U1 ∪ ·· · ∪ Un and y0 ∈ Y . For x ∈ H, let k = min{i ∈
{1, . . . ,n} : x ∈ Ui}. Let us define g : X → Y such that

g(x) =

{
yk if x ∈ H
y0 if x ∈ X \H

By Lemma 4.1, the map g is quasicontinuous and g(Ui) = {yi} for each i ∈
{1, . . . ,n}. �

Theorem 4.10. Let X and Y be topological spaces such that X is Hausdorff.
Then d(Qp(X,Y )) ≤ ω(X) ·d(Y ).

Proof. Let B be a base for X such that |B|≤ ω(X) and U be the family of all
finite pairwise disjoint nonempty members of B. Let D be a dense set in Y such
that |D| ≤ d(Y ) and V be the family of all nonempty finite subsets of D. For
each U = {U1, . . . ,Un} ∈ U and y = {y1, . . . ,yn} ∈ V, by Lemma 4.9, there
exists a quasicontinuous function gU,y : X → Y such that gU,y(Ui) = yi for
each i ∈ {1, . . . ,n}. Then G = {gU,y : U ∈ U,y ∈ V} is dense set in Qp(X,Y )
such that |G| ≤ ω(X) ·d(Y ).

Indeed, for any nonempty basic open set H = [x1, . . . ,xn; V1, . . . ,Vn] in
Qp(X,Y ), there exist U = {U1, . . . ,Un} ∈ U such that xi ∈ Ui and y =
{y1, . . . ,yn}∈V such that yi ∈ Vi for each i ∈{1, . . . ,n}. Thus there is gU,y ∈
G such that gU,y(xi) ∈ Vi for each i ∈{1, . . . ,n} and hence gU,y ∈ H ∩G. �

Corollary 4.11. Let X and Y be topological spaces such that X is second
countable Hausdorff and Y is separable. Then the space Qp(X,Y ) is separable.

Lemma 4.12. Let X and Y be spaces such that X is regular. Then for any
x ∈ X, any nonempty closed set F ⊆ X such that x /∈ F and y1,y2 ∈ Y ,
there exists a quasicontinuous function f : X → Y such that f(x) = y1 and
f(F) = {y2}.

Proof. Since x /∈ F and X is regular, there exist open sets U and V such that
x ∈ U, F ⊆ V and U ∩ V = ∅. Note that x /∈ V . Let us define f : X → Y

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 308



Cardinal invariants and special maps of quasicontinuous functions

such that

f(z) =

{
y1 if z ∈ X \V
y2 if z ∈ V

By Lemma 4.1, f is a quasicontinuous map such that f(x) = y1 and f(F) =
{y2}. �

Theorem 4.13. Let X be a regular space and Y be any nontrivial space. Then
d(X) ≤ ψ(Qp(X,Y )).

Proof. Given a basic open set U = [x1, . . . ,xn; V1, . . . ,Vn] in Qp(X,Y ), let
AU = {x1, . . . ,xn}. Let f0 ∈ Qp(X,Y ) be the constant function such that
f0(x) = y0 for all x ∈ X and γ be a family of open sets with |γ| ≤ ψ(Qp(X,Y ))
such that ∩γ = {f0}. For each G ∈ γ, there exists a basic open set UG =
[xG1 , . . . ,x

G
nG

; V G1 , . . . ,V
G
nG

] such that f0 ∈ UG ⊆ G. We claim that the set
D =

⋃
{AUG : G ∈ γ} is dense in X.

Suppose that x ∈ X \ D and y1 ∈ Y such that y1 6= y0. By Lemma 4.12,
there exists f ∈ Qp(X,Y ) such that f(x) = y1 and f(D) = {y0}. Then f ∈∩γ
and f 6= f0, which is a contradiction. Thus D is dense in X. �

Note that if X is a Tychonoff space, then the result obtained in Theorem
4.13 for Y = R can be concluded from the results d(X) = ψ(Cp(X,R)) [17,
Problem 173] and ψ(Cp(X,R)) ≤ ψ(Qp(X,R)) [17, Problem 159]. We cannot
expect the equality in between d(X) and ψ(Qp(X,R)) even for X = R, because
d(R) = ℵ0, while [8, Example 5.1] shows that ψ(Qp(R,R)) = 2ℵ0 .

Theorem 4.14. Let X be a regular space and Y be a nontrivial T1-space. Then
s(X) ≤ s(Qp(X,Y )).

Proof. Let D be a discrete subspace of X and {Vd : d ∈ D} be a family of
open subsets of X such that Vd ∩D = {d} for each d ∈ D. Choose y1,y2 ∈ Y
such that y1 6= y2, by Lemma 4.12, there exists a quasicontinuous function
fd : X → Y such that fd(d) = y1 and fd(X \ Vd) = {y2}. Then the set
A = {fd : d ∈ D} is discrete in Qp(X,Y ). To see this, choose an open set G
in Y such that y1 ∈ G but y2 /∈ G. Then Ud = [d,G] is open in Qp(X,Y ) and
Ud ∩A = {fd}. Thus s(X) ≤ s(Qp(X,Y )). �

If X is a Tychonoff space and Y = R, then the result obtained in Theorem
4.14 can be obtained from the results s(X) ≤ s(Cp(X,R)) [17, Problem 176]
and s(Cp(X,R)) ≤ s(Qp(X,R)) [17, Problem 159].

Theorem 4.15. Let X and Y be topological spaces such that X is Hausdorff.
Then Q(X,Y ) is dense in Fp(X,Y ).

Proof. Let W = [x1, . . . ,xn; V1, . . . ,Vn] be any nonempty basic open set in
Fp(X,Y ) and f ∈ W . Then f(xi) ∈ Vi for each i ∈ {1, . . . ,n}. Since X is
Hausdorff and x1, . . . ,xn ∈ X, by Lemma 4.2, there exists a quasicontinuous
function g : X → Y such that g(xi) = f(xi) for each i ∈ {1, . . . ,n}. Then
g ∈ W ∩Q(X,Y ). �

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M. Kumar and B. K. Tyagi

Corollary 4.16. Let X be a Hausdorff space and Y be a separable space. Then
the space Qp(X,Y ) has the Souslin property, that is, c(Qp(X,Y )) = ℵ0.

Proof. It is known that if Y is separable then the space Fp(X,Y ) has the
Souslin property [17, Problem 109]. Since Q(X,Y ) is dense in Fp(X,Y ),
c(Qp(X,Y )) = c(Fp(X,Y )) [17, Problem 110]. Thus the space Qp(X,Y ) has
the Souslin property. �

Note that if X is a Tychonoff space then C(X,R) is a dense subset of the
space Qp(X,R) [17, Problem 034]. Also P(R,R), the set of all polynomials
from R to R and U(R,R), the set of all uniformly continuous functions from R
to R are dense subsets of the space Qp(R,R) [17, Problem 041,043].

Proposition 4.17. Let X and Y be topological spaces such that X is Hausdorff
and Y is separable. If F is any locally finite family of nonempty open subsets
of Qp(X,Y ), then F is countable.

Proof. If possible, suppose F is uncountable. Let A be a maximal disjoint
family of nonempty open subsets of Qp(X,Y ) such that each member of A
meets at most finitely many members of F. Because F is locally finite, the set⋃
A is dense in Qp(X,Y ). By Corollary 4.16, c(Qp(X,Y )) ≤ℵ0, which implies

A is countable. Since
⋃
A is dense in Qp(X,Y ), each U ∈ F intersect some

V ∈ A. But every member of A can intersect only finitely many members of
F. Since A is countable, this implies F is countable, a contradiction. �

5. Special maps and the space Qp(X, Y )

The properties of induced maps related to the space C(X,Y ) with the topol-
ogy of pointwise convergence and others are discussed in [14, Chapter II]. Before
discussing the properties of induced maps related to the space Qp(X,Y ), let
us first define these maps in view of quasicontinuous maps.

Note that the composition of two quasicontinuous maps need not be qua-
sicontinuous [16]. However, if f : X → Y is quasicontinuous and g : Y → Z
is continuous, then the composition map gof : X → Z is quasicontinuous. If
g : Y → Z is continuous, then the induced map g∗ : Q(X,Y ) → Q(X,Z)
is defined by g∗(f) = g ◦ f for all f ∈ Q(X,Y ). Also if g ∈ Q(X,Y ), then
the induced map g∗ : C(Y,Z) → Q(X,Z) is defined as g∗(h) = h ◦ g for all
h ∈ C(Y,Z).

Theorem 5.1. For a given continuous map g : Y → Z, the induced map
g∗ : Qp(X,Y ) → Qp(X,Z) such that g∗(f) = g ◦f is continuous. Moreover, if
g is an embedding, then g∗ is also an embedding.

Proof. Let f ∈ Qp(X,Y ) and U be any open set in Qp(X,Z) containing g∗(f).
There is a basic open set V = [x1, . . . ,xn; V1, . . . ,Vn] in Qp(X,Z) such that
g∗(f) ∈ V ⊆ U. Now W = [x1, . . . ,xn; g−1(V1), . . . ,g−1(Vn)] is an open set
in Qp(X,Y ) such that f ∈ W and g∗(W) ⊆ V ⊆ U. Thus the map g∗ is
continuous.

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Cardinal invariants and special maps of quasicontinuous functions

Note that if g is injective then g∗ is also injective. Now to show g∗ :
Qp(X,Y ) → g∗(Qp(X,Y )) is an open map, let [x,V ] be any subbasic open
set in Qp(X,Y ). Since g is an embedding and V is open in Y , there ex-
ists an open set W in Z such that g(V ) = W ∩ g(Y ). We have [x,V ] =
[x,g−1(W)] = g−1∗ ([x,W]). Then g∗([x,V ]) = [x,W]∩g∗(Qp(X,Y )) is open in
g∗(Qp(X,Y )). �

Proposition 5.2. For any space X, there is a continuous map h : Qp(X,R) →
Qp(X,I) such that h(f) = f for each f ∈ Qp(X,I).

Proof. Consider the map g : R → I such that g(t) = −1 if t < −1, g(t) = t if
t ∈ I = [−1, 1] and g(t) = 1 if t > 1. Clearly g is continuous. By Theorem 5.1,
the map h = g∗ : Qp(X,R) → Qp(X,I) defined by h(f) = gof is continuous.
Also h(f) = f for each f ∈ Qp(X,I). �

Theorem 5.3. For a given quasicontinuous map g : X → Y , the map g∗ :
Cp(Y,Z) → Qp(X,Z) such that g∗(h) = h ◦ g is continuous. Moreover, if
g(X) = Y , then g∗ is an embedding.

Proof. Let h0 ∈ Cp(Y,Z) and V = [x1, . . . ,xn; V1, . . . ,Vn] be any basic open
set in Qp(X,Z) containing g

∗(h0). Consider U = [g(x1), . . . ,g(xn); V1, . . . ,Vn]
open in Cp(Y,Z). Then h0 ∈ U and for any h ∈ U, we have g∗(h) ∈ V . Thus
g∗(U) ⊆ V and hence g∗ is continuous.

Now suppose that g(X) = Y . To see g∗ is an injection, let h,h′ ∈ Cp(Y,Z)
such that h 6= h′. Then h(y) 6= h′(y) for some y ∈ Y . Because g(X) =
Y , let x ∈ g−1(y). Then g∗(h)(x) = h(y) 6= h′(y) = g∗(h′)(x). Hence
g∗(h) 6= g∗(h′). To prove g∗ is an embedding, it suffices to show that (g∗)−1 :
g∗(Cp(Y,Z)) → Cp(Y,Z) is continuous. Let g∗(f) ∈ g∗(Cp(Y,Z)) and U =
[y1, . . . ,yn; V1, . . . ,Vn] be any basic open set in Cp(Y,Z) containing f. Choose
xi ∈ g−1(yi) for each i ∈ {1, . . . ,n}. Then V = [x1, . . . ,xn; V1, . . . ,Vn] ∩
g∗(Cp(Y,Z)) is open in g

∗(Cp(Y,Z)) containing g
∗(f). To verify (g∗)−1(V ) ⊆

U, let h ∈ V . Then h = g∗(h′) for some h′ ∈ Cp(Y,Z). Since h = g∗(h′) =
h′ ◦ g ∈ V , we have h′ ◦ g(xi) ∈ Vi for each i ∈ {1, . . . ,n}. This implies
h′(yi) ∈ Vi for each i ∈ {1, . . . ,n} so that h′ ∈ U. Hence h′ = (g∗)−1(h) ∈ U
and we have (g∗)−1(V ) ⊆ U. �

For any space X and maps f,g : X → R such that f is continuous and g
is quasicontinuous, it is easy to see that the map f + g : X → R defined by
(f + g)(x) = f(x) + g(x) is quasicontinuous.

Proposition 5.4. For any space X, the map s : Cp(X,R) × Qp(X,R) →
Qp(X,R) defined by s(f,g) = f + g is continuous.

Proof. Let (f0,g0) ∈ Cp(X,R) ×Qp(X,R) and U be any open set in Qp(X,R)
containing h0 = f0 + g0. There exist x1, . . . ,xn ∈ X and � > 0 such that
h0 ∈ O(h0,x1, . . . ,xn,�) ⊆ U. Then V = O(f0,x1, . . .xn, �2 ) and W =
O(g0,x1, . . .xn,

�
2
) are open in Cp(X,R) and Qp(X,R), respectively. There-

fore V ×W is open in Cp(X,R)×Qp(X,R) containing (f0,g0). We claim that

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M. Kumar and B. K. Tyagi

s(V ×W) ⊆ U. For this, let s(f,g) = f + g ∈ s(V ×W), then |f(xi) + g(xi)−
h0(xi)| ≤ |f(xi) − f0(xi)| + |g(xi) − g0(xi)| < � for all i ∈ {1, . . . ,n}. Thus
f + g ∈ O(h0,x1, . . . ,xn,�) ⊆ U. �

Lemma 5.5. For any x ∈ X, the evaluation map at x, ex : Qp(X,Y ) → Y
defined by ex(f) = f(x) is continuous.

Proof. Let f ∈ Qp(X,Y ) and V be any open set in Y containing f(x). Then
U = [x,V ] is an open set containing f such that ex(U) ⊆ V . Thus ex is
continuous. �

For any space X and A ⊆ X, a family BA of open subsets of X is called a
base at A [5] if each member of BA contains A and for any open set U containing
A, there exists B ∈BA such that B ⊆ U. The character of A in X is defined as
χ(A,X) = ℵ0 + min{|BA| : BA is a base at A}. Note that χ({x},X) = χ(x,X).

Proposition 5.6. Let X be a Hausdorff space. If there exists a compact sub-
space K of the space Qp(X,R) such that χ(K,Qp(X,R)) ≤ ℵ0, then X is
countable.

Proof. Given a basic open set U = [x1, . . . ,xn; V1, . . . ,Vn] in Qp(X,R), let
AU = {x1, . . . ,xn}. Suppose that {Bn : n ∈ N} is a countable base at K
in Qp(X,R). Fix n ∈ N, for each f ∈ K, choose a basic open set Unf such
that f ∈ Unf ⊆ Bn. For open cover {U

n
f : f ∈ K} of K, choose a finite

subcover {Unf1, . . . ,U
n
fmn
} for some mn ∈ N. Let Wn = Unf1 ∪ ·· · ∪ U

n
fmn

and

An = AUn
f1
∪·· ·∪AUn

fmn
, then K ⊆ Wn ⊆ Bn. Clearly A =

⋃
{An : n ∈ N} is

countable. We claim that A = X.
Suppose that x ∈ X \ A. By Lemma 5.5, the map ex : Qp(X,R) → R

defined by ex(f) = f(x) is continuous. Therefore the set ex(K) is bounded in
R. Choose M > 0 such that |f(x)| < M for all f ∈ K. Since W = [x, (−M,M)]
is an open set containing K, there exists k ∈ N such that K ⊆ Bk ⊆ W and
hence Wk = U

k
f1
∪ ·· ·∪Ukfmk ⊆ W . Thus U

k
f1

= [x1, . . . ,xn; V1, . . . ,Vn] ⊆ W
such that x /∈ {x1, . . . ,xn}. Since X is Hausdorff, by Lemma 4.2, choose
g ∈ Qp(X,R) such that g(xi) ∈ Vi for each i ∈{1, . . . ,n} and g(x) = M. Then
g ∈ Wk \W , which is a contradiction. �

The properties of the restriction map related to the space C(X,R) with
the topology of pointwise convergence are discussed in [1]. For Y ⊆ X, the
restriction map is defined as πY : F(X,Z) → F(Y,Z) such that πY (f) = f|Y
for all f ∈ F(X,Z). Note that the restriction of a quasicontinuous map on an
open or a dense subset is quasicontinuous [16]. A map f : X → Y is called
almost onto if f(X) is dense in Y .

Proposition 5.7. Let X be a regular space and Y be an open subset of X. If
the map πY : Q(X,R) → Q(Y,R) such that πY (f) = f|Y is injective, then Y
is dense in X.

Proof. Let h0 ∈ Q(X,R) such that h0(x) = 0 for all x ∈ X. Suppose that πY is
injective but Y is not dense in X so that z ∈ X\Y . By Lemma 4.12, there exists

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Cardinal invariants and special maps of quasicontinuous functions

h ∈ Q(X,R) such that h(z) = 1 and h(Y ) = {0}. We have πY (h) = πY (h0)
but h 6= h0, which is a contradiction. Hence Y is dense in X. �

Theorem 5.8. Let X be a Hausdorff space and Y ⊆ X be open or dense in X.
Then the restriction map πY : Qp(X,R) → Qp(Y,R) such that πY (f) = f|Y is
continuous and almost onto. Moreover, πY is a homeomorphism if and only if
Y = X.

Proof. Consider the natural projection pY : RX → RY such that pY (x) =
x|Y . Then pY is a continuous map [17, Problem 107] and πY = pY |Qp(X,R).
Therefore πY is continuous. By Theorem 4.15, Q(X,R) is dense in RX. Since
pY is continuous, RY = pY (RX) = pY (Qp(X,R)) ⊆ pY (Qp(X,R)). Thus
πY (Qp(X,R)) = pY (Qp(X,R)) is dense in RY and hence also dense in Qp(Y,R).

Now if πY is a homeomorphism and Y 6= X. For x ∈ X\Y , the set D = {f ∈
Qp(X,R) : f(x) = 0} is not dense in Qp(X,R), because D∩[x, (0, 1)] = ∅. But
πY (D) is dense in Qp(Y,R). Let G = [y1, . . . ,yn; V1, . . . ,Vn] be any basic open
set in Qp(Y,R) containing some g. By Lemma 4.2, there exists f ∈ Qp(X,R)
such that f(yi) = g(yi) and f(x) = 0. Then f ∈ D such that πY (f) ∈ G.
Hence πY (D)∩G 6= ∅. Because the image of a dense set πY (D) under the map
(πY )

−1 is D, which is not dense. This implies that (πY )
−1 is not continuous,

which is a contradiction. Finally, if Y = X then πY is the identity map, and
hence a homeomorphism. �

Acknowledgements. The first author acknowledges the fellowship grant of
University Grant Commission, India with Student-ID DEC18-414765. The
authors are thankful to the anonymous referee for his valuable comments and
suggestions.

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