@ Appl. Gen. Topol. 15, no. 2(2014), 147-154doi:10.4995/agt.2014.3181
c© AGT, UPV, 2014

The classical ring of quotients of Cc(X)

Papiya Bhattacharjee a, Michelle L. Knox b and

Warren Wm. McGovern c

a
Penn State Erie, The Behrend College, Erie, PA 16563 (pxb39@psu.edu)

b
Midwestern State University, Wichita Falls, TX 76308 (michelle.knox@mwsu.edu)

c
H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458 (warren.mcgovern@fau.edu)

Abstract

We construct the classical ring of quotients of the algebra of continuous

real-valued functions with countable range. Our construction is a slight

modification of the construction given in [3]. Dowker’s example shows
that the two constructions can be different.

2010 MSC: 54C40, 13B30.

Keywords: ring of continuous functions; ring of quotients; zero-dimensional
space.

1. Introduction

Our aim here is two-fold. We aim to add to the growing knowledge regarding
the ring of continuous functions of countable range on the space X, denoted by
Cc(X), while also supplying a correction to representation of the classical ring
of quotients of Cc(X), denoted qc(X). In this section we supply the relevant
definitions and concepts. In Section 2, we construct qc(X) in the same vein
as the Representation Theorems of Fine, Gillman, and Lambek [2]. The third
section is devoted to studying a specific space which shows why the construction
in Section 2 is needed. We also include an example of Mysior that is peculiar
in its own right.

Throughout this paper, X will denote a zero-dimensional Hausdorff space,
that is, a Hausdorff space with a base of clopen sets. The ring of all real-
valued continuous functions on X is denoted by C(X), and the subring of
C(X) consisting of those functions with countable range is denoted by Cc(X).

Received 23 May 2013 – Accepted 14 November 2013

http://dx.doi.org/10.4995/agt.2014.3181


P. Bhattacharjee, M. L. Knox and W. Wm. McGovern

We may restrict to the class of zero-dimensional spaces because, as it is argued
in [3] and [12], for any space Y there is a zero-dimensional Hausdorff space X
such that Cc(Y ) and Cc(X) are isomorphic as rings.

One of the first results concerning Cc(X) was by W. Rudin [14] who showed
that a compact space X satisfies C(X) = Cc(X) precisely when X is scattered.
In [9] the authors studied general zero-dimensional spaces for which Cc(X) =
C(X), calling such a space functionally countable. Recently, there has been an
interest in Cc(X) as a ring in its own right.

Recall that for f ∈ C(X), Z(f) denotes its zero-set:

Z(f) = {x ∈ X : f(x) = 0}.

The set-theoretic complement of a zero-set is known as a cozero-set and we
label this set by coz(f). We denote the collection of all cozero-sets by coz(X)
and use clop(X) to denote the boolean algebra of clopen subsets of X. When
we consider cozero-sets arising from functions in Cc(X), we get what is denoted
in [6] by clop(X)σ: the set of all countable unions of clopen subsets, i.e. the
set of all σ-clopen sets.

One of the main differences between C(X) and Cc(X) is the realization of the
maximal ideal spaces. The Gelfand-Kolmogorov Theorem states Max(C(X))
is homeomorphic to the Stone-Čech compactification of X, denoted βX. On
the other hand, the maximal ideal space of Cc(X) is homeomorphic to the
Banaschewski compactification of X, denoted β0X. (A proof can be modeled
after Theorem 5.1 of [6].) The most well-known way of realizing βX is as the
collection of all z-ultrafilters of X. A nice way to view β0X is as the Stone
dual of clop(X). In general, βX and β0X are not homeomorphic. The next
theorem characterizing when they are is well-known (see Chapter 6.2 of [1]).

Theorem 1.1. For a zero-dimensional space X the following statements are
equivalent.

(1) βX = β0X
(2) Every cozero-set is a σ-clopen set.
(3) βX is zero-dimensional.

Zero-dimensional spaces X for which βX is zero-dimensional are known as
strongly zero-dimensional spaces. In this short article we are interested in zero-
dimensional spaces which are not strongly zero-dimensional. The third section
deals with one of the most well-known of such examples.

As for references, the text [4] is still pivotal. We also mention [13] and [1]
for topological considerations, definitions, and concepts not explicitly discussed
here. We end this section with a remark about σ-clopens.

Remark 1.2. It should be apparent that if U ∈ clop(X)σ, then there is some
f ∈ Cc(X) such that coz(f) = U. A sketch of this proof is as follows. First

write U =
⋃

n

Kn as a disjoint union of a countable number of clopen sets.

Next, consider the function (call it f) that maps Ki to
1
i
and the rest of X to

0. This function is continuous and is an element of Cc(X). Finally, coz(f) = U.

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The classical ring of quotients of Cc(X)

2. Classical Ring of Quotients

In [12] the authors constructed both the classical ring of quotients and the
maximum ring of quotients of Cc(X). They modeled their construction after
the Representation Theorems of Fine, Gillman, and Lambek [2]. We recall
these after we set up some notation. Let

G(X) = {U ⊆ X : U is a dense open subset of X}

and
G0(X) = {O ⊆ X : O is a dense cozero-set of X}.

We let q(X) and qc(X) denote the classical rings of quotients of C(X) and
Cc(X), respectively. We let Q(X) and Qc(X) denote the maximum rings of
quotients of C(X) and Cc(X), respectively.

Theorem 2.1 ([2]). Suppose X is a Tychonoff space. Then

q(X) = lim
O∈G0(X)

C(O) and Q(X) = lim
U∈G(X)

C(U).

In the above theorem the use of the limit is to describe that the rings are
direct limits of rings of continuous functions. This direct limit can be described
as the union of the rings C(U) modulo the equivalence that f1 ∈ C(U1), f2 ∈
C(U2) are equivalent if they agree on U1 ∩ U2.

In [12] the authors classified qc(X) and Qc(X) for zero-dimensional X as

qc(X) = lim
O∈G0(X)

Cc(O)

and
Qc(X) = lim

U∈G(X)
Cc(U).

Unfortunately, we believe that the classification of qc(X) is incorrect. We now
construct the classical ring of quotients of Cc(X). Let

Gσ(X) = {K ∈ X : K is a dense σ-clopen set of X}.

For the purpose of comparison, we will denote q′(X) = lim
O∈G0(X)

Cc(O).

Theorem 2.2. Let X be a zero-dimensional space. Then

qc(X) = lim
U∈Gσ(X)

Cc(U).

Proof. Given that X ∈ Gσ(X) we have an embedding Cc(X) ≤ qc(X) ≤ q
′(X).

First let f ∈ Cc(X) be a non zero-divisor element of Cc(X). We claim that
coz(f) is a dense subset of X. If not, then X r clX coz(f) is a nonempty
open subset and hence there is a nonempty clopen set of X which is disjoint
from clX coz(f). The characteristic function on said clopen set is a non-zero
function belonging to Cc(X) and annihilates f, a contradiction. Thus, coz(f) ∈
Gσ(X). Restricting f to coz(f) produces an element which is invertible in
Cc(coz(f)) and hence also in qc(X). So every regular element of Cc(X) is

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P. Bhattacharjee, M. L. Knox and W. Wm. McGovern

invertible in qc(X). It follows (by a straightforward ring theoretic argument)
that the classical ring of quotients of Cc(X) is embedded inside of qc(X).

As for the reverse inclusion, the proof follows mutatis mutandi from the proof
of the Representation Theorem of [2] (Theorem 2.6). This was attempted in
Theorem 2.12 of [12]. The only error made there was that when taking a dense
cozeroset U of X there might not be a d ∈ Cc(X) such that coz(d) = U. In fact,
the only change needed from their proof is the modification we have suggested
in using Gσ(X) instead of G0(X). �

Remark 2.3. Observe that qc(X) ≤ q
′(X) since Gσ(X) ⊆ G0(X). One might

question whether both direct limits produce the same rings. In the next section
we will exhibit an example of a zero-dimensional space for which qc(X) < q

′(X).
Of course, if X is strongly zero-dimensional, then G0(X) = Gσ(X) and hence
qc(X) = q

′(X). Remark 3.10 discusses the equality and the question of whether
such a space need be strongly zero-dimensional.

We end this section with some remarks and results whose proofs are in the
same vein as above.

Corollary 2.4. Let F be a proper subfield of R. For a zero-dimensional space

X, the classical ring of quotients of C(X, F) is lim
U∈Gσ(X)

C(U, F). In particular,

the classical ring of quotients of C(X, Q) is q(X, Q) = lim
U∈Gσ(X)

C(U, Q).

3. A Counterexample

We let X be the space defined in 4V of [13]. We give a brief sketch of the
construction. Let W denote the space of countable ordinals equipped with the
interval topology. For an ordinal σ, we use W(σ) to denote the set of ordinals
smaller than σ. Notice that W(ω1) = W.

Let J = R r Q. For x ∈ J, let Jx = {x + r : r ∈ Q} and J = {Jx : x ∈ J}.
Re-index J by J = {Jα : α ∈ W} so that Jα ∩ Jβ = ∅ whenever α 6= β. For
α < ω1, let Uα = Rr

⋃
{Jβ : α < β < ω1}, and let X =

⋃
{{α}×Uα : α < ω1}.

Equip X with the subspace topology from W×R; X is a Tychonoff space. The
space X is similar to Dowker’s Example from [1]. X is a classical example of a
zero-dimensional space that is not strongly zero-dimensional. Another reference
for this space is Exercise 16M of [4].

For notational purposes, let Xσ = {(τ, r) ∈ X : τ < σ}. In other words
Xσ = X ∩ (W(σ) × R). We denote the X-complement of Xσ by X

′

σ and call
such a set a cofinal band of X since X′σ = {(τ, r) ∈ X : σ ≤ τ}.

Since X is not strongly zero-dimensional, we know that not every cozeroset
of X is a σ-clopen. We aim to convince the reader of three things. First, that
if C is a σ-clopen of X, then either C or X r C is a subset of Xσ for some
σ ∈ W. Second, if C is a σ-clopen, then X r clXC is also a σ-clopen. Third,
qc(X) < q

′(X).
We let π : X → W be the continuous projection map. We recall some

subsets of X defined in Section 3 of [8].

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The classical ring of quotients of Cc(X)

Definition 3.1. Let K be a clopen subset of X such that π(K) is cofinal in
W. Define the following sets for r ∈ R and ǫ > 0:

Sǫr = {σ ∈ W : ({σ} × (r − ǫ, r + ǫ)) ∩ X ⊆ K}

and

TK = {r ∈ R : S
ǫ
r is cofinal in W for some ǫ > 0}.

Proposition 3.18 of [8] states TK is an unbounded subset of R. Here we show
more: that TK is an open subset of R.

Proposition 3.2. Suppose K is a clopen subset of X such that π(K) is cofinal
in W. Then

TK = {r ∈ R : there is a σr ∈ W such that {(τ, r) ∈ X : σr ≤ τ} ⊆ K}.

Proof. For r ∈ R notice that by the construction of X, if (σ, r) ∈ X, then for
all σ ≤ τ ∈ W, (τ, r) ∈ X.

If r ∈ TK, then choose σ ∈ W such that (σ, r) ∈ X. Set Y = {(τ, r) ∈ X :
σ ≤ τ} and observe that Y is homeomorphic to W. Let g = χK ∈ C(X) be the
characteristic function on K; K = coz(g). The restriction of g to Y is constant
on a tail (see Chapter 5 [4]), and r ∈ TK, so there is some σr ∈ W such that
for all σr ≤ τ ∈ W, (τ, r) ∈ coz(g) = K.

Next, let r ∈ R have the property that there is σr ∈ W such that {(τ, r) ∈
X : σr ≤ τ} ⊆ K. Assume, by way of contradiction, that r /∈ TK, so for each

n ∈ N we have that S
1

n

r is not cofinal in W. That is, for each n ∈ N there exists
σn ∈ W where ({α} × (r −

1
n
, r + 1

n
)) ∩ X is not a subset of K for all α > σn.

Let σ = sup{σn : n ∈ N}, then for all α > σ, ({α} × (r −
1
n
, r + 1

n
)) ∩ X is not

a subset of K. However, choosing a β ∈ W for which σ, σr ≤ β,then since K
is open there must exist n ∈ N such that ({β} × (r − 1

n
, r + 1

n
)) ∩ X ⊆ K, a

contradiction. Therefore r ∈ TK. �

Proposition 3.3. Let K be a clopen subset of X such that π(K) is cofinal in
W. The set TK is a nonempty open subset of R.

Proof. As we did previously, let g = χK ∈ C(X) with coz(g) = K. Let t ∈ TK,
and suppose, by means of contradiction, that there is no open neighborhood
of t contained in TK. Then there is (without loss of generality) an increasing
sequence of rationals, say {qn}, not belonging to TK which converges to t. Note
that g is eventually zero on a tail of [W × {qn}] ∩ X for each n ∈ N. Thus for
each n ∈ N there is a σn ∈ W such that for all σ > σn we have g((σ, qn)) = 0,
i.e. (σ, qn) /∈ K for every σ > σn.

Let ζ = sup{σn : n ∈ N}. Then consider an appropriate α > ζ such that for
all α ≤ β (β, t) ∈ K. (Such an α exists because t ∈ TK.) But then since K is
open there is a rational qn < t such that (α, qn) ∈ K, a contradiction. �

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P. Bhattacharjee, M. L. Knox and W. Wm. McGovern

Remark 3.4. We observe that the proof of Proposition 3.2 actually shows that
if K is clopen subset of X such that π(K) is cofinal in W, then for any r ∈ R
the set {σ ∈ W : (σ, r) ∈ K} is either bounded by a countable ordinal or
contains a tail of W. In other words,

TK ={r ∈ R : there is a cofinal subset of W, say W, such that (τ, r) ∈ K

for all τ ∈ W}.

This is pivotal in proving our next result.

Proposition 3.5. Let K be a clopen subset of X such that π(K) is cofinal in
W. Then π(X r K) is not cofinal in W. Thus K contains a cofinal band of
X.

Proof. We suppose that both π(K) and π(X r K) are cofinal in W. By the
previous remark it follows that TK ∩ TXrK = ∅ and TK ∪ TXrK = R. By
Proposition 3.3 both TK and TXrK are nonempty open sets. This produces a
disconnection of R, the desired contradiction. �

Proposition 3.6. Let K be a σ-clopen subset of X. Then X r clXK is also
a σ-clopen subset.

Proof. Suppose K is a σ-clopen subset of X.
First consider the case where π(K) is not cofinal in W. Then K ⊆ Xτ for

some τ ∈ W. Xτ is a separable zero-dimensional metrizable space and hence
strongly zero-dimensional. Hence Xτ r clXK is a σ-clopen subset of Xτ. Since
Xτ is a clopen subset of X, it follows that Xτ r clXK is a σ-clopen subset of
X, thus X r clXK = X

′

τ ∪ (Xτ r clXK) is σ-clopen subset of X.
Next consider the case where π(K) is cofinal in W. Then for some τ ∈ W

K contains the cofinal band X′τ . Since X r clXK is an open subset of Xτ . We
mentioned in the previous paragraph that Xτ is a separable zero-dimensional
metrizable space and hence strongly zero-dimensional. It follows that XrclXK
is a σ-clopen subset of Xτ and thus a σ-clopen subset of X. �

Corollary 3.7. Suppose K is a dense σ-clopen subset of X. Then π(K) is
cofinal, and thus K contains a cofinal band of X.

Theorem 3.8. For the space X, qc(X) < q
′(X).

Proof. Let T1 = [W × (−∞, 0)] ∩ X and T2 = [W × (0, ∞)] ∩ X. Both T1
and T2 are cozero-sets of X and hence so is T = T1 ∪ T2. Moreover, T is a
dense subset of X. We note that π(T ) is cofinal in X, but T does not contain
a cofinal band. Therefore, T is not a σ-clopen subset of X. It follows that
T ∈ G0(X) r Gσ(X).

Let f : T → R be defined by

f(x) =

{

1, if x ∈ T1

0, if x ∈ T2.

Then f ∈ Cc(T ) and so f ∈ q
′(X). We claim that f /∈ qc(X). If it were, then

there would exist a dense σ-clopen of X, say V ∈ Gσ(X), and g ∈ Cc(V ) such

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The classical ring of quotients of Cc(X)

that f and g agree on T ∩ V . But since V is a dense σ-clopen set, V contains
a cofinal band of X. Therefore, V ∩ T equals T on this band and so g sends T1
to 1 and T2 to 0. But g is defined on the whole band, contradicting continuity
at points of the form (τ, 0) for large enough τ. �

Remark 3.9. Proposition 3.6 is interesting on its own. The proposition yields
for a zero-dimensional space Y , both rings qc(Y ) and limU∈Gσ(Y ) C(U) are
von Neumann regular rings. The proof would be modeled after the proofs of
Proposition 1.2 [10] and Theorem 1.3 of [7]. Simply, you would need that X is
σ-clopen complemented.

The ring limU∈Gσ(X) C(U) has not been studied except in the case that X
is strongly zero-dimensional. We conjecture that the ring can be realized as
the classical ring of quotients of the Alexandroff Algebra A(X) (see [6] or [5]
for more information).

Remark 3.10. Let X∗ denote the finer topology on R2 defined by Mysior [11]
using D = Q × Q. X∗ is another example of a zero-dimensional space that
is not strongly zero-dimensional. The countable set D is precisely the set of
all isolated points of X∗. Thus, D is the smallest dense σ-clopen subset of
X∗. It follows that qc(X

∗) = q′(X∗) = C(N) = q(X∗) = Q(X∗). Therefore,
in general it is not the case that the equality qc(X) = q

′(X) forces X to be
strongly zero-dimensional. We are unable to characterize in any nice way when
qc(X) = q

′(X).

Acknowledgements. We would like to express our gratitude to the referee
for his/her helpful comments.

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