@
Appl. Gen. Topol. 23, no. 2 (2022), 281-286

doi:10.4995/agt.2022.17080

© AGT, UPV, 2022

The largest topological ring of functions

endowed with the m-topology

Tarun Kumar Chauhan and Varun Jindal

Department of Mathematics, Malaviya National Institute of Technology Jaipur, Jaipur-302017,

Rajasthan, India (rajputtarun.chauhan@gmail.com, vjindal.maths@mnit.ac.in)

Communicated by A. Tamariz-Mascarúa

Abstract

The purpose of this article is to identify the largest subring of the ring
of all real valued functions on a Tychonoff space X, which forms a
topological ring endowed with the m-topology.

2020 MSC: 54C30; 54C40; 54H13.

Keywords: locally bounded functions; real valued functions; rings of func-
tions; m-topology.

1. Introduction

For a Tychonoff (completely regular and Hausdorff) space X, RX represents
the ring of all real valued functions defined on X. Two of its important sub-
rings are C(X), the set of all continuous functions in RX, and C∗(X), the
set of all bounded and continuous members of RX. The algebraic properties
of the rings C(X) and C∗(X) vis-à-vis topological properties of X have been
studied extensively in the literature (see, [3]). Besides studying the algebraic
properties of C(X), one can also define many interesting topologies on C(X).
Consequently, one may study the interaction between various algebraic struc-
tures of C(X) with the corresponding topology on it. Two commonly studied
topologies on C(X) and C∗(X) are the u-topology (uniform topology) and the
m-topology. Both of these topologies can be defined on RX (definitions are
given in the next section).

Received 27 January 2022 – Accepted 14 July 2022

http://dx.doi.org/10.4995/agt.2022.17080


T. K. Chauhan and V. Jindal

Though the convergence concerning u-topology, popularly known as the uni-
form convergence, has been known for centuries, the m-topology was introduced
by E. Hewitt ([5]) in 1948. In many aspects, the u-topology is the most rel-
evant for studying C∗(X), while the m-topology is appropriate to study the
ring C(X) (see, Theorem 1 and Theorem 3 in [5]). The m-topology has been
studied in detail in [1, 2, 4, 6, 7, 8, 11, 12]. For more about u-topology and
m-topology, we refer readers to the recent research monograph [13].

It is known that C∗(X) equipped with the u-topology forms a topological
ring while C(X) need not. However, C(X) equipped with the m-topology
is a topological ring. But RX with any of these topologies does not form
a topological ring in general. There are several interesting subrings that are
intermediate between C∗(X) and RX, such as the ring of all Baire one functions
and the ring of all locally bounded functions. The main results of the paper
(Theorems 2.8 and 2.11) help to recognize such subrings, which are topological
rings endowed with u-topology and m-topology. It can be shown that C∗(X) is
the largest subring of C(X), which is a topological ring under the u-topology.
It follows that a subring S(X) of C(X) is a topological ring for the u-topology if
and only if S(X) ⊆ C∗(X). This article aims to formulate the largest subrings
of RX, which form topological rings when equipped with u-topology and m-
topology, respectively.

2. Main Results

Throughout this article, X is assumed to be a Tychonoff space (though
we may specify that it has some additional properties). We first recall the
definitions of u-topology and m-topology.

Definition 2.1. The u-topology or uniform topology (denoted by τu) is de-
termined on RX by taking all sets of the form

Bu(f,�) = {g ∈ RX : |f(x) −g(x)| < �,∀x ∈ X}; (� > 0 is constant)
as a base for the neighborhood system at f ∈ RX.

Definition 2.2. The m-topology (denoted by τm) is determined on RX by
taking all sets of the form

Bm(f,η) = {g ∈ RX : |f(x) −g(x)| < η(x),∀x ∈ X}; (η ∈ U+(X))
as a base for the neighborhood system at f ∈ RX. Here U+(X) represents the
set of all positive units in C(X).

Clearly, τm is finer than τu. These topologies coincide if and only if X is
pseudocompact.

The u-topology can also be determined on RX by using the positive units
of C∗(X) in a way given in the following proposition.

Proposition 2.3. Let τ be the topology on RX determined by taking all sets
of the form

B(f,γ) = {g ∈ RX : |f(x) −g(x)| < γ(x),∀x ∈ X}; (γ ∈ U∗+(X))

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 282



The largest topological ring of functions

as a base for the neighborhood system at f for each f ∈ RX, where U∗+(X)
denotes the set of all positive units in C∗(X). Then τ = τu on RX.

There is a considerable difference between the units of the rings C(X) and
C∗(X). A function f ∈ C(X) is a positive unit of C(X) if and only if f(x) > 0.
But a function f ∈ C∗(X) is a positive unit of C∗(X) if and only if f(x) > 0
and 1

f
∈ C∗(X). Equivalently, f ∈ C∗(X) is a positive unit of C∗(X) if and

only if inf{f(x) : x ∈ X} > 0. So it is clear that every positive unit of C∗(X)
is also a positive unit of C(X), but the converse need not be true.

To identify the largest subrings of RX, which are topological rings endowed
with τu and τm, we define two families B(X) and D(X) of functions in the
following manner.

B(X) = {f ∈ RX : ∃ ψ ∈ U∗+(X), |f(x)| < ψ(x) for all x ∈ X},
D(X) = {f ∈ RX : ∃ φ ∈ U+(X), |f(x)| < φ(x) for all x ∈ X}.

It is not hard to see that B(X) and D(X) are subrings of RX, and B(X) is
the same as the family of all bounded functions in RX. Clearly, C(X) ⊆D(X)
and B(X) ⊆D(X). Also D(X) = B(X) if and only if X is pseudocompact.

We now relate D(X) with two important subrings of RX, namely B1(X) and
LB(X) which denote respectively, the ring of all Baire one functions and the
ring of all locally bounded functions. Recall that a function f : X → R is called
Baire one if f is the pointwise limit of a sequence of continuous functions from
X to R, and f is called locally bounded if it is bounded on some neighborhood
of each point of X.

Clearly, D(X) ⊆ LB(X). In general, this containment is strict. To see
an example of a space X for which D(X) 6= LB(X), we need the following
definition and discussion.

Definition 2.4 (Horne, [9]). A space X is called a cb-space if for each h ∈
LB(X), there exists f ∈ C(X) such that |h| ≤ f.

The next proposition follows immediately from the Definition 2.4.

Proposition 2.5. D(X) = LB(X) if and only if X is a cb-space.

It is known that a cb-space is countably paracompact, and a normal space
is a cb-space if and only if it is countably paracompact ([10]). Consequently,
for a non countably paracompact space X, D(X) 6= LB(X).

The following example proves that B1(X) and D(X) may not be comparable.

Example 2.6. Let X = R with the usual topology. Define the function f :
R → R such that f(x) = 0 for all x ∈ (−∞, 0] and f(x) = 1

x
for all x ∈ (0,∞).

For each n ∈ N, define a function fn : R → R such that

fn(x) =




0 if x ∈ (−∞, 0],
n2x if x ∈ (0, 1/n),
1
x

if x ∈ [1/n,∞).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 283



T. K. Chauhan and V. Jindal

Clearly, each fn is continuous and the sequence (fn) converges pointwise to f.
Therefore, f ∈ B1(X). But f /∈D(X). Now define another function h : R → R
such that h(x) = 0 for every rational number x and h(x) = 1 for every irrational
number x. We can easily see that h ∈D(X) \B1(X).

Using Proposition 2.5 and Example 2.6, by Theorem 2.8, we can conclude
that LB(X) and B1(X) need not form topological rings endowed with the
m-topology.

Theorem 2.7. D(X) endowed with τm is a topological ring.

Proof. The continuity of the map (f,g) → f + g is easy to check. We only
prove that the map (f,g) → fg is continuous. Let f,g ∈ D(X). So there
exist φf,φg ∈ U+(X) such that |f(x)| < φf (x) and |g(x)| < φg(x) for all
x ∈ X. Let Bm(fg,η) be any basic neighborhood of fg in (D(X),τm) for some
η ∈ U+(X). Consider the basic neighborhoods Bm(f,η1) and Bm(g,η2) of f
and g respectively for η1 =

η
2(1+φg)

and η2 =
η

2(φf +η1+1)
. It is enough to show

that for any h1 ∈ Bm(f,η1) and h2 ∈ Bm(g,η2), we have h1h2 ∈ Bm(fg,η). It
follows as

|(fg)(x) − (h1h2)(x)| ≤ |g(x)||f(x) −h1(x)| + |h1(x)||g(x) −h2(x)|
< φg(x)η1(x) + |h1(x)|η2(x) for all x ∈ X
< η(x) for all x ∈ X.

�

Our next theorem establish the fact that D(X) is the largest subring of RX
which is a topological ring endowed with the m-topology.

Theorem 2.8. Let S(X) be a subring of RX. Then the following conditions
are equivalent:

(a) S(X) endowed with τm is a topological ring;
(b) S(X) ⊆D(X).

Proof. (a) ⇒ (b). Suppose S(X) * D(X). Let f ∈ S(X) \D(X). We show
that pointwise multiplication (f,g) → fg is not continuous at point (0X,f),
where 0X is the constant function zero on X. Consider the basic neighborhood
Bm(0X, 1) of the function 0Xf = 0X in (S(X),τ

m). Since f /∈D(X), for every
η ∈ U+(X) there exists a point xη ∈ X such that |f(xη)| ≥

2

η(xη)
, that is,∣∣∣∣η(xη)2 f(xη)

∣∣∣∣ ≥ 1. Therefore for any η,µ ∈ U+(X), we have η2 ∈ Bm(0X,η)
and f ∈ Bm(f,µ) but

η

2
f /∈ Bm(0X, 1).

(b) ⇒ (a). It follows from Theorem 2.7. �

Corollary 2.9. LB(X) equipped with τm is a topological ring if and only if X
is a cb-space.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 284



The largest topological ring of functions

Corollary 2.10. For a first countable space X, the following conditions are
equivalent:

(a) X is discrete;
(b) RX = D(X);
(c) RX endowed with τm is a topological ring;
(d) RX = LB(X).

Proof. The implication (a) ⇒ (b) is immediate and the implications (b) ⇒
(c) ⇒ (d) follow from Theorem 2.8.

(d) ⇒ (a). Suppose there is a non-isolated point x0 ∈ X. Since X is first
countable, there exists a sequence (xn) of distinct points in X \ {x0} which
converges to x0. Define a function f : X → R such that f(xn) = n for every
n ∈ N and f(x) = 0 for all x ∈ X \ {xn : n ∈ N}. It is easy to see that
f /∈ LB(X). We arrive at a contradiction. �

It should be noted that X being first countable is used only to prove the
implication (d) ⇒ (a). It may be interesting to know whether Corollary 2.10
is true for any Tychonoff space X.

Theorem 2.11. B(X) is the largest subring of RX, which is a topological ring
endowed with τu.

Proof. It can be proved in a manner similar to Theorems 2.7 and 2.8. �

Corollary 2.12. RX equipped with τu is a topological ring if and only if X is
finite.

Corollary 2.13. Every subring of RX which forms a topological ring under τu
is also a topological ring under τm.

Corollary 2.14. For a space X, the following conditions are equivalent:

(a) LB(X) endowed with τu is a topological ring;
(b) B(X) = LB(X);
(c) X is a pseudocompact cb-space;
(d) X is countably compact.

Proof. The equivalence (a) ⇔ (b) follows from Theorem 2.11 and (c) ⇔ (d)
follows from Theorem 9 of [10].

(b) ⇔ (c). It follows from Proposition 2.5 and the fact that D(X) = B(X)
if and only if X is pseudocompact. �

We conclude this article with the following question.

Question 2.15. For what spaces X, the ring B1(X) endowed with τ
m or τu

forms a topological ring?

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 285



T. K. Chauhan and V. Jindal

Acknowledgements. We thank the referee for his/her valuable suggestions.
The second author acknowledges the support of NBHM Research Grant 02011/
6/2020/NBHM(R.P) R&D II/6277.

References

[1] F. Azarpanah, F. Manshoor and R. Mohamadian, Connectedness and compactness in

C(X) with m-topology and generalized m-topology, Topol. Appl. 159 (2012), 3486–3493.
[2] F. Azarpanah, M. Paimann and A. R. Salehi, Connectedness of some rings of quotients

of C(X) with the m-topology, Comment. Math. Univ. Carolin. 56 (2015), 63–76.

[3] L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, New York,
1976. Reprint of the 1960 edition, Graduate Texts in Mathematics, No. 43

[4] J. Gómez-Pérez and W. W. McGovern, The m-topology on Cm(X) revisited, Topol.

Appl. 153, no. 11 (2006), 1838–1848.
[5] E. Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64

(1948), 45–99.

[6] L. Holá and V. Jindal, On graph and fine topologies, Topol. Proc. 49 (2017), 65–73.
[7] L. Holá and R. A. McCoy, Compactness in the fine and related topologies, Topol. Appl.

109 (2001), 183–190.

[8] L. Holá and B. Novotný, Topology of uniform convergence and m-topology on C(X),
Mediterr. J. Math. 14 (2017): 70.

[9] J. G. Horne, Countable paracompactness and cb-spaces, Notices. Amer. Math. Soc. 6
(1959), 629–630.

[10] J. Mack, On a class of countably paracompact spaces, Proc. Amer. Math. Soc. 16 (1965),

467–472.
[11] G. Di Maio, L. Holá, D. Holý and R. A. McCoy, Topologies on the space of continuous

functions, Topol. Appl. 86 (1998), 105–122.

[12] R. A. McCoy, Fine topology on function spaces, Int. J. Math. Math. Sci. 9 (1986),
417–424.

[13] R. A. McCoy, S. Kundu and V. Jindal, Function spaces with uniform, fine and graph

topologies, Springer Briefs in Mathematics, Springer, Cham, 2018.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 286