CaoGreenAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 7, No. 2, 2006

pp. 253-264

The ideal generated by σ-nowhere
dense sets

Jiling Cao and Sina Greenwood
∗

Abstract. In this paper, we consider the ideal Iσ generated by
all σ-nowhere dense sets in a topological space. Properties of this ideal

and its relations with the Volterra property are explored. We show that

Iσ is compatible with the topology for any given topological space, an
analogue to the Banach category theorem. Some applications of this

result and the Banach category theorem are also given.

2000 AMS Classification: 26A15, 28A05, 54C05, 54E52.

Keywords: Compatible, Ideal, Resolvable, σ-nowhere dense, Volterra, Weakly
Volterra.

1. Introduction

Let (X, τ ) be a topological space. An ideal I on X is a family of subsets
of X such that (i) B ∈ I , if B ⊂ A and A ∈ I ; (ii) A ∪ B ∈ I , if A, B ∈ I .
If (ii) is replaced by (ii)’

⋃

n<ω
An ∈ I for any sequence 〈An : n < ω〉 in I ,

then I is called a σ-ideal. For any given ideal I on X, the minimal σ-ideal
containing I shall be called the σ-extension of I . An ideal is said to be proper
if it is not equal to the power set P(X) of X. All these notions come from
the algebra of P(X) if some appropriate operations are introduced. Ideals in
general topological spaces were considered in [12], and a more modern study
can be found in [7].

One connection between an ideal and the topology on a given topological
space arises through the concept of the local function of a subset with respect
to the ideal.

∗The first author was supported by the Foundation for Research, Science and Technology
of New Zealand under project number UOAX0240.



254 J. Cao and S. Greenwood

Definition 1.1 ([3, 7]). Let (X, τ ) be a topological space, and let I be an ideal
on X.

A∗(I ) = {x ∈ X : A ∩ N 6∈ I for every N ∈ N (x)}

is called the local function of A ⊂ X with respect to I , where N (x) denotes
the collection of all neighbourhoods of x in (X, τ ).

The local function operator was used in [3] in the investigation of ideal
resolvability. Observe that cl∗(A) = A ∪ A∗(I ) defines a Kuratowski closure
operator, which generates a new topology τ ∗(I ) on X finer than τ . It can
be easily checked that B(I ) = {U r I : U ∈ τ and I ∈ I } is a base for
the topology τ ∗(I ). For general properties of the local function operator and
τ ∗(I ), we refer readers to [7].

Ideals have been frequently used in fields closely related to topology, such as,
real analysis, measure theory, and descriptive set theory. The following ideals
have been of particular interest:

In – the ideal of all nowhere dense sets in (X, τ ),
Im – the σ-ideal of all meager sets in (X, τ ),
Ib – some σ-ideal consisting of boundary sets in (X, τ ),
I0 – the σ-ideal of all Lebesgue measure zero sets in R

n.

For example, by using Ib, Semadeni [14] established a purely topological gen-
eralization of the Carathéodory characterization of functions which are equal
to Riemann-integrable functions almost everywhere. The crucial fact that Se-
madeni required is the following: If a set A is locally in Ib (that is, for each
x ∈ A, there is a neighbourhood of x in the subspace A which is a member of
Ib) then A is a member of Ib. Requirements similar to this one have been
used in many other places. For instance, in Bourbaki’s integration theory in
locally compact spaces, it is required that a set which is locally negligible is of
measure zero. Another interesting result is that if a set is locally in Im then
it is a member of Im. This is the Banach category theorem, first proved by
Banach for metric spaces in [1].

In [2] and [5], Cao, Gauld, Greenwood and Piotrowski studied the Volterra
property. The class of Volterra spaces is closely related to the class of Baire
spaces. Cao and Gauld [2] proved an analogue to the Oxtoby’s Banach cat-
egory theorem stated in [9], namely, the union of any family of non-weakly
Volterra open subspaces is still non-weakly Volterra. Since the ideal Im plays
an important role in the study of Baire and other related properties, and in
particular the Banach category theorem can be formulated by using the σ-ideal
Im, it is natural to consider which ideal might play a similar role in the study
of the Volterra property, and if there exists a result analogous to the Banach
category theorem for the Volterra property in terms of ideals.

In the following we show the ideal Iσ is such an ideal. After discussing some
basic properties of Iσ in Section 2, relations between the Volterra property and
Iσ are investigated in Section 3, and the compatibility of Iσ with the topology



The ideal generated by σ-nowhere dense sets 255

of any given topological space is established in Section 4. In the last section,
we shall consider some applications of the Banach category theorem and its
analogue.

2. Preliminaries

In this section, we discuss some basic properties of the ideal generated by
all σ-nowhere dense sets in a topological space, where σ-nowhere dense sets are
defined as follows.

Definition 2.1. A subset of a topological space (X, τ ) is called σ-nowhere dense
if it is an Fσ-set with empty interior.

Note that any subset with empty interior is also called a boundary set. In
general, the family of σ-nowhere dense sets in a topological space (X, τ ) is not
an ideal. The smallest ideal on X that contains all σ-nowhere dense sets in
(X, τ ) will be denoted by Iσ. Sometimes, Iσ is also named as the ideal gen-
erated by the family of σ-nowhere dense sets. It is clear that In ⊂ Iσ ⊂ Im,
and that the σ-extension of Iσ is precisely Im. The following two examples
show that Iσ may be distinct from In and Im in a general topological space
(X, τ ), and the topology τ ∗(Iσ) may be strictly between τ and the discrete
topology on X.

Example 2.2. Let (X, τ ) be the real line R with the usual topology. Since
Q ∈ Iσ rIn, we have In 6= Iσ. It follows that τ

∗(Iσ) 6= τ , since Q is τ
∗(Iσ)-

closed, but not τ -closed. Suppose that {x} is τ ∗(Iσ)-open for some x ∈ X,
then there exists an open set U ∈ τ and an I ∈ Iσ such that {x} = U r I. But
I is a subset of a countable union of nowhere dense sets, and since X is Baire,
this gives a contradiction. Hence, we conclude that τ ∗(Iσ) is not the discrete
topology on X. �

Example 2.3. If X is any countably infinite set with the cofinite topology,
then any open set is meager but not in Iσ. �

In Example 3.8 below, we shall give a Tychonoff space (X, τ ) in which Iσ 6=
Im, and in which τ

∗(Iσ) is neither τ nor discrete.

Lemma 2.4. Let (X, τ ) be a topological space, and m ∈ N. For each family
{Ei : i < m} of σ-nowhere dense sets, there is another family {Gi : i < m} of
σ-nowhere dense sets such that Gi ⊂ Ei for each i < m,

⋃

i<m
Ei =

⋃

i<m
Gi,

and for each k < m, intτ (
⋃

i≤k
Gi) ∩ intτ (

⋃

i≥k
Gi) = ∅.

Proof. Let G0 = E0. For each k < m, let Nk = intτ (
⋃

i≤k
Ei) ∩ intτ (

⋃

i≥k
Ei),

and Gi = Ei r
⋃

k<i
Nk for each 0 < i < m (Note that N0 = ∅, and so

G1 = E1). Observe that for each i < m, intτ Gi ⊂ intτ Ei = ∅, and since
⋃

k<i
Nk is open, Gi is an Fσ-set in X.

Suppose that x ∈
⋃

i<m
Ei r

⋃

i<m
Gi. Let n = min{i : x ∈ Ei}. Then x ∈

En and hence x ∈
⋃

k<n
Nk. Suppose x ∈ Nj and j < n, then x ∈ intτ

⋃

i≤j
Ei



256 J. Cao and S. Greenwood

contradicting the minimality of n. Thus
⋃

i<m
Ei =

⋃

i<m
Gi. Then, it remains

to establish that for each k < m,

intτ

(

⋃

i≤k
Gi

)

∩ intτ

(

⋃

i≥k
Gi

)

= ∅.

To this end, we shall show that

intτ

(

⋃

i≤k
Gi

)

∩ intτ

(

⋃

i≥k
Gi

)

⊂ Gk

for all k < m. Suppose that x ∈ intτ (
⋃

i≤k
Gi) ∩ intτ (

⋃

i≥k
Gi). Observe that

x 6∈
⋃

i>k
Gi because for each i > k,

Gi ⊂ Ei r Nk ⊂ Ei r
(

intτ

(

⋃

j≤k
Gj

)

∩ intτ

(

⋃

j≥k
Gj

))

.

It follows from x ∈ intτ (
⋃

i≥k
Gi) that x ∈ Gk. �

The following theorem and its corollaries are useful in the sequel.

Theorem 2.5. In a topological space (X, τ ), any subset of X, that is the union
of finitely many σ-nowhere dense sets, can be expressed as the union of exactly
two σ-nowhere dense subsets.

Proof. Suppose that A =
⋃

i<m
(
⋃

n<ω
C(n, i)), where each C(n, i) is closed,

and intτ (
⋃

n<ω
C(n, i)) = ∅. For each i < m, let Ei =

⋃

n<ω
C(n, i). We will

define two σ-nowhere dense sets, D0 and D1, such that A = D0 ∪ D1. By
Lemma 2.4, we may assume that for each k < m,

intτ

(

⋃

i≤k
Ei

)

∩ intτ

(

⋃

i≥k
Ei

)

= ∅.

Now, define D0 ⊂ X and D1 ⊂ X by

D0 = E0 ∪ (E1 r intτ (E0 ∪ E1)) ∪ ... ∪
(

Em−1 r intτ
(
⋃

i<m Ei
))

,

and

D1 = Em−1 ∪ (Em−2 r intτ (Em−1 ∪ Em−2)) ∪ ... ∪
(

E0 r intτ
(
⋃

i<m
Ei
))

.

Suppose that x ∈ A. If x ∈ Em−1, then x ∈ D1. Now, suppose that x ∈
(
⋃

i<m−1
Ei) r D0. Then there exists some k < m − 1 such that x ∈ Ek, and

hence x ∈ intτ

(

⋃

i≤k Ei

)

. Thus x 6∈ intτ

(

⋃

i≥k Ei

)

, and so x ∈ D1. We have



The ideal generated by σ-nowhere dense sets 257

shown that A = D0 ∪ D1. Observe that

D0 =
⋃

n<ω

{

(E0 ∩ C(n, 0)) ∪ ((E1 r intτ (E0 ∪ E1)) ∩ C(n, 1))

∪... ∪

((

Em−1 r intτ

(

⋃

i<m

Ei

))

∩ C(n, m − 1)

)}

=
⋃

n<ω

{

C(n, 0) ∪ (C(n, 1) r intτ (E0 ∪ E1))

∪... ∪

(

C(n, m − 1) r intτ

(

⋃

i<m

Ei

))}

,

and hence D0, and similarly D1, is an Fσ-set of (X, τ ). In addition, suppose
that W = intτ D0 6= ∅. Then, we let

k = min
{

j < m : W ⊂ intτ

(

⋃

i≤j
Ei

)}

.

For each y ∈ W and l ≥ k, y 6∈ El r intτ (
⋃

i≤l
Ei) and hence there exists jy < k

such that y ∈ Ejy r intτ (
⋃

i≤jy
Ei). Hence W ⊂ intτ (

⋃

i<k
Ei), contradicting

the minimality of k. This implies that intτ D0 = ∅. In a similar way, one
can show intτ D1 = ∅. Therefore, D0 and D1 are σ-nowhere dense sets of
(X, τ ). �

Corollary 2.6. For any topological space (X, τ ), I ∈ Iσ if and only if there
are two σ-nowhere dense sets E and F in X such that I ⊂ E ∪ F .

Given a topological space (X, τ ), and a subspace Y ⊂ X, let τY denote
the subspace topology on Y , and let Iσ(Y ) denote the ideal generated by all
σ-nowhere dense sets in the subspace (Y, τY ).

Lemma 2.7. Let (X, τ ) be a topological space, and let O ⊂ X be an open
subspace. Then Iσ(O) = Iσ ∩ P(O).

Proof. If I ∈ Iσ(O), then by Corollary 2.6, there are two σ-nowhere dense
subsets E and F in the subspace O such that I ⊂ E ∪ F . Suppose that
E =

⋃

n<ω
En and F =

⋃

n<ω
Fn, where intOE = intOF = ∅, and for each

n < ω, En and Fn are closed subsets of the subspace (O, τO). Then, it is easy to
check that

⋃

n<ω
clτ En ∈ Iσ and

⋃

n<ω
clτ Fn ∈ Iσ, and hence I ∈ Iσ ∩P(O),

and so Iσ(O) ⊂ Iσ ∩ P(O).

Conversely, if I ∈ Iσ ∩ P(O), then by Corollary 2.6, I ⊂ O and I ⊂ E ∪ F
for two σ-nowhere dense sets in (X, τ ). Furthermore, E ∩ O and F ∩ O are two
σ-nowhere dense sets in the subspace (O, τO ). Hence, I ∈ Iσ(O), and thus
Iσ ∩ P(O) ⊂ Iσ(O). �



258 J. Cao and S. Greenwood

3. The Volterra property and Iσ

In this section we explore relationships between the Volterra property and
the ideal Iσ of a topological space (X, τ ).

Definition 3.1 ([5]). A topological space is called Volterra (resp. weakly
Volterra) if the intersection of every two dense Gδ-sets is dense (resp. non-
empty).

As a consequence of Theorem 2.5, we obtain Theorem 2.3 of [2].

Theorem 3.2 ([2]). A topological space (X, τ ) is weakly Volterra if and only
if the intersection of any finitely many dense Gδ-sets is non-empty.

Proposition 3.3. Let (X, τ ) be a topological space. Then Iσ is proper if and
only if (X, τ ) is weakly Volterra.

Proof. By Corollary 2.6, X ∈ Iσ if and only if there exist σ-nowhere dense
sets D and E such that X = D ∪ E, if and only if (X r D) ∩ (X r E) = ∅, if
and only if X is not weakly Volterra. �

Lemma 3.4 ([5]). A topological space (X, τ ) is Volterra if and only if Iσ does
not contain a non-empty open subset of (X, τ ).

Corollary 3.5. Let (X, τ ) be a topological space. If A ⊂ X and A∗(Iσ ) = X,
then A is a Volterra subspace of (X, τ ).

Corollary 3.6. Let (X, τ ) be a topological space. If there exists a subset A ⊂ X
such that A∗(Iσ) = X then (X, τ ) is Volterra.

Recall that a space is resolvable if it contains two disjoint dense subsets,
and a space is irresolvable if it is not resolvable. A strongly irresolvable space
is a space all of whose open subspaces are irresolvable. The notion of I -
resolvability, was introduced in [3], and is defined as follows: suppose I is an
ideal on X, a topological space (X, τ ) is I -resolvable if there are two disjoint
subsets A, B of X such that A∗(I ) = B∗(I ) = X. Observe that by definition,
if a space is I -resolvable for some ideal I then it is resolvable, and if (X, τ ) is
I -resolvable then it is J -resolvable for any ideal J ⊂ I . In general, there
is no relationship between the Volterra property and resolvability, but it turns
out that the stronger property, I -resolvability or strong irresolvability has an
interesting relationship with the Volterra property.

The proof of the following proposition is easy, and so is omitted.

Proposition 3.7. If a topological space (X, τ ) is either strongly irresolvable or
Iσ-resolvable, then it is Volterra.

In general, the converse of Proposition 3.7 is not true.

Example 3.8. First, note that R with the usual topology is Volterra, but is
resolvable. Next, let X be any set with |X| = ℵ0. Van Douwen showed, in [4,
Example 1.9], that there exists a Tychonoff topology τ on X such that



The ideal generated by σ-nowhere dense sets 259

• (X, τ ) is dense-in-itself;
• (X, τ ) is hereditarily irresolvable, that is, every subspace of (X, τ ) is irre-

solvable; and
• there exists a nowhere dense set which is not closed in (X, τ ).

Thus (X, τ ) is not Iσ-resolvable, and by Proposition 3.7, (X, τ ) is a Volterra
space.

Note also that Iσ 6= Im since by Proposition 3.3, Iσ is proper, but since
X is Tychonoff, dense-in-itself and countable, Im is not proper. Furthermore,
τ ∗(Im) 6= τ

∗(Iσ). It is clear that τ
∗(Im) is the discrete topology on X,

whereas τ ∗(Iσ ) is not. We show in Corollary 5.4 in Section 5 that τ
∗(Iσ) is

not even regular. �

It was shown in [3, Theorem 3.3] that a space is Im-resolvable if and only if
it has two disjoint dense Baire subspaces. In the light of this result, it is inter-
esting to consider the relationship between Iσ-resolvability and the Volterra
property. Since the relationship between Baire spaces and Im resembles that
between Volterra and Iσ, we might expect an analogous result. However, we
have the following proposition.

Proposition 3.9. If a topological space (X, τ ) is Iσ-resolvable, then it contains
two disjoint dense Volterra subspaces.

Proof. Suppose that A and B are two disjoint subsets of X such that A∗(Iσ) =
B∗(Iσ) = X. It is clear that A and B are dense in X. By Corollary 3.5, A
and B are two Volterra subspaces of (X, τ ). �

The converse of Proposition 3.9 is not in general true.

Example 3.10 ([5]). Let E and O be the sets of all even positive integers
and all odd positive integers respectively, and let X = E ∪ O. For any pair
m, n ∈ X, we define Umn ⊂ X by letting

Umn = {x : x ∈ E and x ≥ 2m; or x ∈ O and x ≥ 2n − 1}.

It can easily be checked that τ = {∅} ∪ {Umn : m, n ∈ X} is a topology on X.
Now, the topological space (X, τ ) enjoys the following three properties:

• E and O are two disjoint dense Volterra subspaces of X.
• The space X is not weakly Volterra.
• The space X is not Iσ-resolvable. This follows from Proposition 3.7. �

4. Compatibility of Iσ and its topology

In this section we establish the compatibility of Iσ and τ for any topological
space (X, τ ).

Definition 4.1. Let (X, τ ) be a topological space. An ideal I on X is said
to be compatible with τ , denoted by I ∼ τ , if for each A ⊂ X and for every
point x ∈ A there is a neighborhood U of x such that U ∩ A ∈ I , then A ∈ I .



260 J. Cao and S. Greenwood

It is easy to see that In ∼ τ for any topological space (X, τ ). The next
theorem is called the Banach category theorem in the literature, see [12].

Theorem 4.2 ([12]). For any topological space (X, τ ), Im ∼ τ .

Several equivalent versions and generalizations via ideals of Theorem 4.2 are
given in [11]. Also, in [11, Example 26], it is shown that there is a topological
space (X, τ ) and an ideal between In and Im on X which is not compatible
with τ . However, Iσ is a compatible ideal as shown below.

Theorem 4.3. Let (X, τ ) be a topological space. Then Iσ ∼ τ .

Proof. Suppose A ⊂ X, and for each x ∈ A there is an open neighborhood Ux
of x such that A ∩ Ux ∈ Iσ. Let ℑ be the set of all collections of non-empty
open subsets of X with the following two properties:

(i) each collection V ∈ ℑ is pairwise disjoint;
(ii) For each collection V ∈ ℑ and each member V ∈ V , there exists some

x ∈ A such that V ⊂ Ux.

By Zorn’s lemma, there is a maximal collection V ∈ ℑ. Decompose A as

A = (A ∩ (
⋃

V )) ∪ (A r
⋃

V ).

Since {Ux : x ∈ A} is an open cover of A, by the maximality of V , we have

A r
⋃

V ⊂
⋃

V r
⋃

V . Furthermore, following from the fact
⋃

V r
⋃

V ∈ In ⊂ Iσ,

we have A r
⋃

V ∈ Iσ. Thus, in order to show A ∈ Iσ, it will suffice to show
A ∩ (

⋃

V ) ∈ Iσ.

For each V ∈ V , A ∩ V ⊂ A ∩ Ux for some x ∈ A. Since A ∩ Ux ∈ Iσ, then
A∩V ∈ Iσ. By Lemma 2.7, A∩V ∈ Iσ(V )∩Iσ(

⋃

V ). Now, for each V ∈ V ,
by Corollary 2.6, there are closed sets D(V, n, i) in the subspace (V, τV ) such
that for each i < 2, intV

(
⋃

n<ω D(V, n, i)
)

= ∅ and

A ∩ V ⊂
⋃

i<2

(
⋃

n<ω
D(V, n, i)

)

.

Since V is a pairwise disjoint family of non-empty open sets, each D(V, n, i) is
closed in the subspace

⋃

V , the set D(n, i) =
⋃

{D(V, n, i) : V ∈ V } is also
closed in the subspace

⋃

V . Thus, for each i < 2,
⋃

n<ω
D(n, i) is an Fσ-set

in
⋃

V . For any fixed i < 2, suppose that there is some non-empty open
subset W of

⋃

V such that W ⊂
⋃

n<ω
D(n, i). Then there exists some V ∈ V

such that W ∩ V 6= ∅ and W ∩ V ⊂
⋃

n<ω
D(V, n, i). This is a contradiction,

because
⋃

n<ω
D(V, n, i) has empty interior in the subspace V . Therefore, for

each i < 2,
⋃

n<ω D(n, i) is σ-nowhere dense in
⋃

V . Furthermore, following

from the fact that A ∩ (
⋃

V ) ⊂
(
⋃

n<ω
D(n, 0)

)

∪
(
⋃

n<ω
D(n, 1)

)

, we obtain
A ∩ (

⋃

V ) ∈ Iσ(
⋃

V ). Finally, by Lemma 2.7, A ∩ (
⋃

V ) ∈ Iσ. Therefore,
A ∈ Iσ. �

Corollary 4.4 ([2]). In any topological space (X, τ ), the union of any family
of open non-weakly Volterra subspaces is not weakly Volterra.



The ideal generated by σ-nowhere dense sets 261

Proof. Let V be a family of open non-weakly Volterra subspaces of (X, τ ). For
any x ∈

⋃

V , select Vx ∈ V such that x ∈ Vx. Moreover, it can shown easily
that an open set G is not a weakly Volterra subspace if and only if G ∈ Iσ.
So, Vx is an open neighborhood of x such that Vx ∩ (

⋃

V ) ∈ Iσ. Therefore,
by Theorem 4.3,

⋃

V ∈ Iσ. This implies that
⋃

V is not a weakly Volterra
subspace of (X, τ ). �

5. Applications

In this section, we shall give some applications of the Banach Category
Theorem and Theorem 4.3.

5.1. Feeble continuity of the inversion of a para-topological group.
Recall that a group with a topology is called a paratopological group if its
multiplication is jointly continuous. In addition, if its inversion is also contin-
uous, then it is called a topological group. The Sorgenfrey line is an example
of a paratopological group which is not a topological group. Bouziad observed
that if the inversion of a paratopological group is quasi-continuous, then it is
a topological group (see [8, Lemma 4]). The problem arising here is whether
the inversion of a paratopological group can have some weak form of continu-
ity under certain circumstances. For example, Guran [6] asked the following
question:

Problem 5.1 ([6]). Let (G, ·, τ ) be a Baire regular paratopological group.
Must the inversion of (G, ·, τ ) be feebly continuous?

Recall that a mapping f : (X, τ ) → (Y, µ) is feebly continuous if for every
non-empty open set V ⊂ Y , intτ (f

−1(V )) 6= ∅. The general answer to Ques-
tion 5.1 is negative as shown by Ravsky in [13]. Next, we shall show that the
answer is affirmative under some mild restriction. To this end, we need to recall
a definition. A family N of non-empty subsets in a topological space (X, τ ) is
called a π-network of (X, τ ) if for each non-empty open set U ∈ τ , there exists
some N ∈ N such that N ⊂ U .

Proposition 5.2. Let (G, ·, τ ) be a Baire paratopological group. If (G, τ ) has
a countable π-network, the inversion of (G, ·, τ ) is feebly continuous.

Proof. Let U ⊂ G be a non-empty open set. Pick a point y ∈ U . Since
the multiplication is jointly continuous at (e, y), one can choose open sets V
and W in (G, τ ) such that e ∈ V , y ∈ W and V · W ⊂ U . It follows that
W −1 · V −1 ⊂ U −1. Now, observe that (W −1)∗(Im) is a closed set of G. Let
g ∈ (W −1)∗(Im). Since g · V is an open set containing g, by the construction
of (W −1)∗(Im), (g · V ) ∩ W

−1 6∈ Im. In particular, (g · V ) ∩ W
−1 6= ∅,

which implies g ∈ W −1 · V −1. Thus, (W −1)∗(Im) ⊂ W
−1 · V −1 ⊂ U −1. Let

N = {Nn : n < ω} be a countable π-network for (G, τ ), and let

Gn = {h ∈ G : Nn ⊂ h · W }

for each n < ω. It can easily be checked that G =
⋃

n<ω
Gn. Since G is Baire,

there exists n0 < ω such that Gn0 6∈ Im. Select any point g0 ∈ Nn0 . Then,



262 J. Cao and S. Greenwood

Gn0 ⊂ g0 · W
−1. It follows that g0 · W

−1 6∈ Im and hence, W
−1 6∈ Im. On the

other hand, by the Banach category theorem, W −1 ∩(Gr(W −1)∗(Im)) ∈ Im.
Thus, we can conclude that (W −1)∗(Im) 6∈ Im, otherwise, W

−1 ∈ Im, which
gives a contradiction. Consequently,

∅ 6= intτ ((W
−1)∗(Im)) ⊂ intτ (W

−1 · V −1) ⊂ intτ (U
−1),

and therefore the inversion of G is feebly continuous. �

5.2. Pointwise Iσ-continuity. Let X, Y be topological spaces, and I an
ideal on X. According to [10], a mapping f : X → Y is called pointwise I -
continuous, if for every x ∈ X and every neighborhood V of f (x), there exists
a neighborhood U of x in X such that U r f −1(V ) ∈ I .

Proposition 5.3. Let X be a topological space, and Y a regular topological
space. Then a mapping f : X → Y is pointwise Iσ-continuous if and only if
f|X0 : X0 → Y is continuous for some closed X0 ⊂ X with X r X0 ∈ Iσ.

Proof. Sufficiency is in [10, Theorem 2]. For necessity, let

X0 = X r
⋃

{G : G is open in X and G ∈ Iσ}.

By Theorem 4.3, X0 ⊂ X is closed and X r X0 ∈ Iσ. Furthermore, by [10,
Lemma 4], f|X0 : X0 → Y is continuous. �

Corollary 5.4. Let (X, τ ) be a Volterra space. If the topology τ ∗(Iσ) is regular
then τ ∗(Iσ) = τ .

Proof. Let f : (X, τ ) → (X, τ ∗(Iσ)) be the identity mapping. It is clear that f
is pointwise Iσ-continuous. By Proposition 5.3, there is a closed set X0 ⊂ X
such that X r X0 ∈ Iσ. Since (X, τ ) is Volterra, by Lemma 3.4, X r X0 = ∅.
Thus f is continuous, which implies that τ ∗(Iσ ) = τ . �

5.3. Continuity of additive mappings. Let (X, τ ) be a topological space.
A subset A ⊂ X is said to be Iσ-modulo to a subset B ⊂ X provided that
(A r B) ∪ (B r A) ∈ Iσ. Let

M (Iσ) = {A ⊂ X : A is Iσ-modulo to some open set of X}.

It can be easily checked that M (Iσ) satisfies the following properties:

• τ ⊂ M (Iσ);
• if A ∈ M (Iσ), then X r A ∈ M (Iσ);
• if A ⊂ X is closed, then A ∈ M (Iσ); and
• if A, B ∈ M (Iσ), then A ∪ B ∈ M (Iσ).

Proposition 5.5. Let f : X → Y be an additive mapping from a linear
topological space X into a linear topological space Y , and let A ⊂ X. If
A ∈ M (Iσ) r Iσ and f|A : A → Y is continuous, then f : X → Y is
continuous.



The ideal generated by σ-nowhere dense sets 263

Proof. By the additivity of f , it will suffice to show that f is continuous at
0 in X. To do this, let V be an arbitrary neighborhood of 0 in Y . Choose
an open neighborhood W of 0 in Y such that W − W ⊂ V . It is clear that
f −1(W ) − f −1(W ) ⊂ f −1(V ). Since A 6∈ Iσ, from Theorem 4.3, there must
be a point x0 ∈ A such that A ∩ N 6∈ Iσ for every open neighborhood N
of x0. Without loss of generality, we may assume that x0 = 0. Now, as
f|A : A → Y is continuous at 0, there is an open neighborhood O of 0 in X
such that f (A ∩ O) ⊂ W . Put B = A ∩ O. Then B ∈ M (Iσ) r Iσ.

Next, we show that B − B is a neighborhood of 0 in X. To this end, let
U ⊂ X be an open subset with (B r U ) ∪ (U r B) ∈ Iσ. Since B 6∈ Iσ, U
must be non-empty, and thus U − U is a non-empty open neighborhood of 0.
Now, take an arbitrary point x ∈ U −U . Then (x + U )∩U is a non-empty open
subset of X. Moreover, we claim that (x + U ) ∩ U 6∈ Iσ, otherwise (x + U ) ∩ U
would be a weakly Volterra subspace of X. As

X =
⋃

{n · ((x + U ) ∩ U ) : n ∈ N},

by Corollary 4.4, the space X itself is not weakly Volterra. From Proposi-
tion 3.3, Iσ is not proper, otherwise B ∈ Iσ. This is a contradiction, verifying
the claim. In addition, from the previous claim and since

(U r B) ∪ ((x + U ) r (x + B)) ∈ Iσ,

and

((x + U ) ∩ U ) r ((x + B) ∩ B) ⊂ (U r B) ∪ ((x + U ) r (x + B)),

we conclude that (x+ B)∩B 6= ∅, and thus x ∈ B −B. Hence, U −U ⊂ B −B,
which implies that B − B is a neighborhood of 0 in X.

Finally, since B − B ⊂ f −1(W ) − f −1(W ) ⊂ f −1(V ), we conclude that
f −1(V ) is a neighborhood of 0 in X. Hence, f is continuous at 0. �

References

[1] S. Banach, Théorème sur les ensembles de première catégorie, Fund. Math. 16 (1930),
395–398.

[2] J. Cao and D. Gauld, Volterra spaces revisited, J. Austral. Math. Soc. 79 (2005), 61–76.
[3] J. Dontchev, M. Ganster and D. Rose, Ideal resolvability, Topology Appl. 93 (1999),

1–16.
[4] E. K. van Douwen, Applications of maximal topologies, Topology Appl. 51 (1993), 125–

139.
[5] D. Gauld, S. Greenwood and Z. Piotrowski, On Volterra spaces III: topological opera-

tions, Topology Proc. 23 (1998), 167–182.
[6] I. Guran, Cardinal invariants of paratopological groups, 2nd International Algebraic

Conference in Ukraine, Kyiv, Vinnytsia, 1999.
[7] D. Janković and T. Hamlett, New topologies from old via ideals, Amer. Math. Monthly

97 (1990), 295–310.
[8] P. Kenderov, I. Kortezov and W. Moors, Topological games and topological groups,

Topology Appl. 109 (2001), 157–165.
[9] R. Haworth and R. McCoy, Baire spaces, Dissertationes Math. 141, 1977.



264 J. Cao and S. Greenwood

[10] J. Kaniewski and Z. Piotrowski, Concerning continuity apart from a meager set, Proc.
Amer. Math. Soc. 98 (1986), 324–328.

[11] J. Kaniewski, Z. Piotrowski and D. Rose, Ideal Banach category theorems, Rocky Moun-
tain J. Math. 28 (1998), 237–251.

[12] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
[13] O. Ravsky, Paratopological groups II, Math. Stud. 17 (2002), 93–101.
[14] Z. Semadeni, Functions with sets of points of discontinuity belonging to a fixed ideal,

Fund. Math. 52 (1963), 26–39.

Received June 2005

Accepted January 2006

Jiling Cao (cao@math.auckland.ac.nz)
Department of Mathematics, The University of Auckland, Private Bag 92019,
Auckland, New Zealand
Current address: School of Mathematical Sciences, Auckland University of
Technology, Private Bag 92006, Auckland 1020, New Zealand
Email: jiling.cao@aut.ac.nz

Sina Greenwood (sina@math.auckland.ac.nz)
Department of Mathematics, The University of Auckland, Private Bag 92019,
Auckland, New Zealand