On I-Baire spaces

TLDR: In this article, the concept of I-Baire spaces is introduced and characterizations and properties of these spaces are given, and it is shown that (X; ) is Baire if and only if (X;; I) is I-baire for any ideal I on X.

Abstract: In this paper, the concept of I-Baire spaces is introduced, and characterizations and properties of these spaces are given. It is shown that (X; ) is Baire if and only if (X;; I) is I-Baire for any ideal I on X.

Full text

Filomat 27:2 (2013), 301–310
DOI 10.2298/FIL1302301L
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
On I-Baire spaces
Zhaowen Li
a
, Funing Lin
a
a
School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, P.R.China
Abstract. In this paper, the concept of I-Baire spaces is introduced, and characterizations and properties
of these spaces are given. It is shown that (X, τ) is Baire if and only if (X, τ, I) is I-Baire for any ideal I on
X.
1. Introduction
It is well known that a Baire space is defined as a space in which every countable intersection of dense
open subsets is dense, or equivalently a space with the property that every nonempty open subspace is
nonmeager. Baire spaces have various applications in complete metric spaces. To develop applications of
Baire spaces, some researchers have studied some spaces such as hyperspaces, Volterra spaces (see [10, 19]).
Recently, Chakrabarti and Dasgupta [2] have investigated Baire spaces with minimal structure.
Ideals on topological spaces were studied by Kuratowski [17] and Vaidyanathaswamy [22]. Their
applications have been investigated intensively (see [3, 6, 7, 13, 16, 18, 20]).
The aim of this paper is to introduce and study I-Baire spaces. Some characterizations and properties
of I-Baire spaces, including their subspaces, are investigated. Finally, some mapping theorems and a
topological sum theorem on I-Baire spaces are discussed.
2. Preliminaries
Let X be a nonempty set, let 2
X
be a family of all subsets of X and let I ⊂ 2
X
. I is called an ideal (resp.
a σ-ideal) on X, if it satisfies the following conditions:
(1) If A ∈ I and B ⊂ A, then B ∈ I;
(2) If A, B ∈ I (resp. {A
n
: n ∈ N} ⊂ I), then A ∪ B ∈ I (resp.

n∈N
A
n
∈ I).
If τ is a topology on X and I is an ideal on X, then (X, τ, I) is called an ideal topological space or simply
an ideal space.
Let (X, τ, I) be an ideal space. An operator (·)
∗
: 2
X
−→ 2
X
, called a local function [17] of A with respect
to τ and I, is defined as follows: for any A ⊂ X,
A
∗
(I, τ) = {x ∈ X : U ∩ A < I for every U ∈ τ(x)}
2010 Mathematics Subject Classification. Primary 54A05; Secondary 54A10, 54C08, 54E52
Keywords. Ideals, Baire spaces, I-Baire spaces, ∗-dense sets, nowhere ∗-dense sets, ∗-first category sets, topological sum
Received: 03 April 2012; Revised: 11 October 2012; Accepted: 15 October 2012
Communicated by Ljubi
ˇ
sa D.R. Ko
ˇ
cinac
Research supported by the National Natural Science Foundation of China (No. 11061004, 10971186, 71140004) and the Science
Research Project of Guangxi University for Nationalities (No. 2011QD015).
Email addresses: lizhaowen8846@126.com (Zhaowen Li), linfuning1016@163.com (Funing Lin)

Z. Li, F. Lin / Filomat 27:2 (2013), 301–310 302
where τ(x) = {U ∈ τ : x ∈ U}.
An operator cl
∗
(·) : 2
X
−→ 2
X
is defined as follows: for any A ⊂ X,
cl
∗
(A)(I, τ) = A ∪ A
∗
(I, τ).
Since cl
∗
(·) is a Kuratowski closure operator, cl(·)
∗
generates a topology τ
∗
(I, τ), called ∗-topology. It is
easy to prove that τ
∗
(I, τ) ⊃ τ.
When there is no chance for confusion, we will simply write τ
∗
for τ
∗
(I, τ), A
∗
for A
∗
(I, τ), c
∗
A for
cl
∗
(A)(I, τ) and i
∗
A for int
∗
(A)(I, τ), where
int
∗
(A)(I, τ) = X − cl
∗
(X − A)(I, τ).
A ⊂ X is called ∗-closed [16] if c
∗
A = A, and A is called ∗-open (i.e., A ∈ τ
∗
) if X − A is ∗-closed. Obviously,
A is ∗-open if and only if i
∗
A = A.
Throughout this paper, spaces always mean topological spaces or ideal spaces on which no separation
axiom is assumed, and all mappings are onto. Sometimes, (X, τ) and (X, τ, I) are simply written by X. N
denotes the set of all natural integers. Let U ⊂ 2
X
, A ⊂ X and x ∈ X. U
A
denotes {U

A : U ∈ U} and U(x)
denotes {U ∈ U : x ∈ U}. The closure of A and the interior of A are denoted by cA and iA respectively, and
we have
iA ⊂ i
∗
A ⊂ A ⊂ c
∗
A ⊂ cA.
Let (X, τ, I) be an ideal space and let Y ⊂ X. Then (Y, τ
Y
, I
Y
) is an ideal space, where τ
Y
= {U ∩ Y : U ∈ τ}
and I
Y
= {I ∩ Y : I ∈ I} = {I ∈ I : I ⊂ Y}. For a space (X, τ, I) (resp. (X, τ
∗
, I)) with A ⊂ Y ⊂ X, the closure of
A and the interior of A in the subspace (Y, τ
Y
, I
Y
) (resp. (Y, τ
∗
Y
, I
Y
)) are denoted by c
Y
A and i
Y
A (resp. c
∗
Y
A
and i
∗
Y
A), respectively.
Given A ⊂ X and some operators γ
i
: 2
X
−→ 2
X
(i = 1, 2, · · · , n). For convenience, we simply denote
γ
1
(γ
2
(· · · (γ
n
(A)) · · · )) by γ
1
γ
2
· · · γ
n
A.
Lemma 2.1. ([12]) Let (X, τ, I) be an ideal space and let A ⊂ X. If U ∈ τ, then U ∩ c
∗
A ⊂ c
∗
(U ∩ A).
Lemma 2.2. ([12]) Let (X, τ, I) be an ideal space and let A ⊂ Y ⊂ X. Then c
∗
Y
(A) = c
∗
A ∩ Y.
3. ∗-denseness and nowhere ∗-denseness
Definition 3.1. A subset A of an ideal space (X, τ, I) is called
(1) ∗-dense [11], if c
∗
A = X;
(2) nowhere ∗-dense [1], if i
∗
cA = ∅.
The family of all nowhere ∗-dense subsets of an ideal space X shall be denoted by N
∗
(X) or simply by
N
∗
when no ambiguity is present.
Remark 3.2. Let (X, τ, I) be an ideal space and let B ⊂ A ⊂ X.
(1) If A is ∗-dense in X, then A is dense in X.
(2) If A is nowhere ∗-dense in X, then A is nowhere dense in X.
(3) If A ∈ N
∗
, then B ∈ N
∗
.
(4) A ∈ N
∗
if and only if cA ∈ N
∗
.
Proposition 3.3. Let (X, τ, I) and (X, τ, J) be two ideal spaces with I ⊂ J and A ⊂ X. If A is ∗-dense in (X, τ, J),
then A is ∗-dense in (X, τ, I).
Proof. This follows from the fact that I ⊂ J implies A
∗
(J, τ) ⊂ A
∗
(I, τ).
Proposition 3.4. Let (X, τ, I) and (X, σ, I) be two ideal spaces with τ ⊂ σ and A ⊂ X. If A is ∗-dense in (X, σ, I),
then A is ∗-dense in (X, τ, I).

Z. Li, F. Lin / Filomat 27:2 (2013), 301–310 303
Proof. This follows from the fact that τ ⊂ σ implies A
∗
(I, σ) ⊂ A
∗
(I, τ).
Proposition 3.5. Let (X, τ, I) be an ideal space. Then A ⊂ X is ∗-dense in X if and only if U ∩ A , ∅ for any
U ∈ τ
∗
− {∅}.
Proof. Necessity. Let A be ∗-dense in X and let U ∈ τ
∗
− {∅}. Pick x ∈ U. Then x ∈ X = c
∗
A = A ∪ A
∗
.
Case 1. x ∈ A.
Then x ∈ U ∩ A. So U ∩ A , ∅.
Case 2. x ∈ A
∗
.
Suppose U ∩ A = ∅. Since X − U is ∗-closed in X, (X − U)
∗
⊂ X − U. Then U ⊂ X − (X − U)
∗
. By x ∈ U,
x < (X − U)
∗
. It follows that V ∩ (X − U) ∈ I for some V ∈ τ(x). By U ∩ A = ∅, A ⊂ X − U. This implies
V ∩ A ⊂ V ∩ (X − U). Then V ∩ A ∈ I. So x < A
∗
, a contradiction. Thus, U ∩ A , ∅.
Sufficiency. Suppose c
∗
A , X. Put U = X − c
∗
A. Then U ∈ τ
∗
− {∅}. But U ∩ A = (X − c
∗
A) ∩ A = ∅. This is
a contradiction.
Proposition 3.6. Let (X, τ, I) be an ideal space and let A ⊂ X. The following are equivalent.
(1) A ∈ N
∗
;
(2) For each U ∈ τ
∗
− {∅}, U 1 cA;
(3) For each U ∈ τ
∗
− {∅}, U − cA ∈ τ
∗
− {∅};
(4) X − cA is ∗-dense in X.
Proof. (1)=⇒(2) Suppose that U ⊂ cA for some U ∈ τ
∗
− {∅}. Since A ∈ N
∗
, we have U = i
∗
U ⊂ i
∗
cA = ∅.
Thus, U = ∅, a contradiction.
(2)=⇒(3) Let U ∈ τ
∗
− {∅}. By (2), U 1 cA. Then U − cA , ∅. Since X − cA ∈ τ and τ ⊂ τ
∗
, we have
X − cA ∈ τ
∗
. Note that U ∈ τ
∗
. Thus U − cA = U ∩ (X − cA) ∈ τ
∗
− {∅}.
(3)=⇒(4) Let U ∈ τ
∗
− {∅}. By (3), U − cA ∈ τ
∗
− {∅}. This implies that U ∩ (X − cA) = U − cA , ∅ for any
U ∈ τ
∗
− {∅}. By Proposition 3.5, X − cA is ∗-dense in X.
(4)=⇒(1) Let X − cA be ∗-dense in X. Then X = c
∗
(X − cA) = X − i
∗
cA. This implies i
∗
cA = ∅ and thus
A ∈ N
∗
.
Proposition 3.7. Let (X, τ, I) be an ideal space and let A ⊂ Y ⊂ X.
(1) If A ∈ N
∗
(Y), then A ∈ N
∗
(X).
(2) If Y ∈ τ
∗
and A ∈ N
∗
(X), then A ∈ N
∗
(Y).
(3) If Y is ∗-dense in X and A ∈ N
∗
(X), then A ∈ N
∗
(Y).
Proof. (1) Let A ∈ N
∗
(Y). By Proposition 3.6, Y − c
Y
A is ∗-dense in Y. So Y ⊂ c
∗
(Y − c
Y
A) = c
∗
(Y − cA ∩ Y) =
c
∗
(Y − cA). Since X − cA = (Y − cA) ∪ (X − Y), we have c
∗
(X − cA) = c
∗
(Y − cA) ∪ c
∗
(X − Y) ⊃ Y ∪ (X − Y) = X.
Then X − cA is ∗-dense in X. By Proposition 3.6, A ∈ N
∗
(X).
(2) Let Y ∈ τ
∗
and A ∈ N
∗
(X). For any W ∈ τ
∗
Y
− {∅}, W = U ∩ Y for some U ∈ τ
∗
− {∅}. Note that U, Y ∈ τ
∗
.
Then W ∈ τ
∗
. Since A ∈ N
∗
(X), by Proposition 3.6, X−cA is ∗-dense in X. By Proposition 3.5, (X−cA)∩W , ∅.
Note that (Y − c
Y
A) ∩ W = (Y − cA ∩ Y) ∩ W = (Y − cA) ∩ W = ((X − cA) ∩ Y) ∩ (U ∩ Y) = (X − cA) ∩ W. By
Proposition 3.5, Y − c
Y
A is ∗-dense in Y. By Proposition 3.6, A ∈ N
∗
(Y).
(3) Let Y be ∗-dense in X and A ∈ N
∗
(X). Since A ∈ N
∗
(X), by Proposition 3.6, X − cA is ∗-dense in
X. For any W ∈ τ
∗
Y
− {∅}, W = U ∩ Y for some U ∈ τ
∗
− {∅}. By Proposition 3.5, (X − cA) ∩ U , ∅. Then
(X − cA) ∩ U ∈ τ
∗
− {∅}. Note that Y is ∗-dense in X. By Proposition 3.5, (Y − c
Y
A) ∩ W = (Y − cA ∩ Y) ∩ W =
(Y − cA) ∩ W = ((X − cA) ∩ Y) ∩ (U ∩ Y) = Y ∩ ((X − cA) ∩ U) , ∅. By Proposition 3.5, Y − c
Y
A is ∗-dense in
Y. By Proposition 3.6, A ∈ N
∗
(Y).
Proposition 3.8. Let (X, τ, I) be an ideal space and let A, B ⊂ X. If A, B ∈ N
∗
, then A ∪ B ∈ N
∗
.
Proof. Since A, B ∈ N
∗
, i
∗
cA = i
∗
cB = ∅. Note that i
∗
c(A ∪ B) = X − c
∗
(X − c(A ∪ B)) = X − c
∗
(X − (cA) ∪ (cB)) =
X − c
∗
((X − cA) ∩ (X − cB)). Since X − cA ∈ τ, by Lemma 2.1, we have X − c
∗
((X − cA) ∩ (X − cB)) ⊂
X − (X − cA)∩ c
∗
(X − cB) = cA ∪ i
∗
cB = cA∪ ∅ = cA. So i
∗
c(A∪ B) ⊂ cA. Then, i
∗
c(A∪ B) = i
∗
i
∗
c(A∪ B) ⊂ i
∗
cA = ∅.
Thus A ∪ B ∈ N
∗
.

Z. Li, F. Lin / Filomat 27:2 (2013), 301–310 304
Theorem 3.9. Let (X, τ, I) be an ideal space. Then N
∗
is an ideal on X.
Proof. This holds by Remark 3.2 (3) and Proposition 3.8.
4. ∗-first category and ∗-second category
Recall that a set in a Baire space is said to be a first category set or a meager set, if it can be written as a
countable union of nowhere dense sets. We shall also give an analogous notion for I-Baire spaces.
Definition 4.1. Let (X, τ, I) be an ideal space and let A ⊂ X.
(1) A is called ∗-first category or ∗-meager in X, if there exists a sequence {A
n
} consisting of nowhere ∗-dense
subsets of X such that A =

n∈N
A
n
.
(2) A is called ∗-second category or ∗-nonmeager in X, if A is not ∗-first category in X.
(3) A is called ∗-residual or ∗-comeager in X, if X − A is ∗-first category in X.
The family of all ∗-first category subsets (resp. all first category subsets) of an ideal space X shall be
denoted by M
∗
(X) or simply by M
∗
(resp. M(X) or M) when no ambiguity is present.
Remark 4.2. For any ideal space, N
∗
⊂ M
∗
⊂ M.
Proposition 4.3. Let (X, τ, I) be an ideal space and let A ⊂ Y ⊂ X.
(1) If A ∈ M
∗
(Y), then A ∈ M
∗
(X).
(2) If Y ∈ τ
∗
and A ∈ M
∗
(X), then A ∈ M
∗
(Y).
(3) If Y is ∗-dense in X and A ∈ M
∗
(X), then A ∈ M
∗
(Y).
Proof. These hold by Proposition 3.7.
Corollary 4.4. Let (X, τ, I) be an ideal space and let A ⊂ Y ⊂ X.
(1) If A is ∗-second category in X, then A is ∗-second category in Y.
(2) If Y ∈ τ
∗
and A is ∗-second category in Y, then A is ∗-second category in X.
(3) If Y is ∗-dense in X and A is ∗-second category in Y, then A is ∗-second category in X.
Proof. These hold by Proposition 4.3.
Proposition 4.5. Let (X, τ, I) be an ideal space and let A ⊂ Y ⊂ X. If Y ∈ M
∗
, then A ∈ M
∗
.
Proof. Let Y ∈ M
∗
and A ⊂ Y. Then Y =

n∈N
Y
n
where Y
n
∈ N
∗
for each n ∈ N. So A = A ∩ Y = A ∩ (

n∈N
Y
n
) =

n∈N
(A ∩ Y
n
). Put A
n
= A ∩ Y
n
for each n ∈ N. Note that each A
n
⊂ Y
n
. By Remark 3.2 (3), A
n
∈ N
∗
.
Consequently, A =

n∈N
A
n
∈ M
∗
.
Corollary 4.6. Let (X, τ, I) be an ideal space and let A ⊂ Y ⊂ X. If A is ∗-second category in X, then Y is ∗-second
category in X.
Proof. This holds by Proposition 4.5.
Proposition 4.7. Let (X, τ, I) be an ideal space. If F
n
∈ M
∗
for each n ∈ N, then

n∈N
F
n
∈ M
∗
.
Proof. This is obvious.
Theorem 4.8. Let (X, τ, I) be an ideal space. Then M
∗
is a σ-ideal on X.
Proof. This holds by Proposition 4.5 and 4.7.

Z. Li, F. Lin / Filomat 27:2 (2013), 301–310 305
5. I -Baire spaces
5.1. The concept of I-Baire spaces
Definition 5.1. Let (X, τ, I) be an ideal space. X is called I-Baire, if for any sequence {G
n
} consisting of open
and ∗-dense subsets of X,

n∈N
G
n
is dense in X.
Example 5.2. Let X = N, A = {1, 3, 5, · · · }, B = X − A,
τ = {∅} ∪ {A ∪ M : M ∈ 2
B
} and I = 2
B
.
It is easily proved that (X, τ, I) is an ideal space.
Let {G
n
} be any sequence consisting of open and ∗-dense subsets of X. We have

n∈N
G
n
⊃ A. Note that
cA = X. Then c(

n∈N
G
n
) = X. Thus (X, τ, I) is I-Baire.
Theorem 5.3. Let (X, τ, I) and (X, τ, J) be two ideal spaces with I ⊂ J. If (X, τ, I) is I-Baire, then (X, τ, J) is
J-Baire.
Proof. This holds by Proposition 3.3.
Theorem 5.4. Let (X, τ, I) be an ideal space. If (X, τ
∗
) is Baire, then (X, τ, I) is I-Baire.
Proof. Let {G
n
} be a sequence of open and ∗-dense subsets of (X, τ, I). Note that τ ⊂ τ
∗
, and for each n ∈ N,
G
n
is ∗-dense in (X, τ) if and only if G
n
is dense in (X, τ
∗
). Since (X, τ
∗
) is Baire, c
∗
(

n∈N
G
n
) = X and thus
c(

n∈N
G
n
) = X. Hence (X, τ, I) is I-Baire.
Theorem 5.5. Let (X, τ, I) be an ideal space. The following are equivalent.
(1) X is I-Baire;
(2) Each nonempty ∗-residual subset A of X is dense in X;
(3) Each U ∈ τ − {∅} is ∗-second category in X;
(4) M
∗
⊂ 2
X
− (τ − {∅});
(5) iF = ∅ for each F ∈ M
∗
.
Proof. (1)=⇒(2) Suppose that A is ∗-residual in X. Then X − A =

n∈N
A
n
where A
n
∈ N
∗
. By Remark 3.2 (4),
cA
n
∈ N
∗
for each n ∈ N. By Proposition 3.6, each X − cA
n
∈ τ is ∗-dense in X. Now
A = X − (X − A) = X −

n∈N
A
n
=

n∈N
(X − A
n
) ⊃

n∈N
(X − cA
n
).
Since X is I-Baire, c(

n∈N
(X − cA
n
)) = X. Then cA = X. Thus A is dense in X.
(2)=⇒(3) Suppose that U is not ∗-second category in X for some U ∈ τ − {∅}. Then U ∈ M
∗
.
Case 1. Suppose U , X. Since U ∈ M
∗
, by (2), X − U is ∗-residual in X and then X − U is dense in X.
Note that U ∈ τ − {∅}. Then (X − U) ∩ U , ∅. This is a contradiction.
Case 2. Suppose U = X. By Proposition 4.5, V ∈ M
∗
(X) for any open set V ⊂ U. Now it satisfies the
condition of Case 1 and so we omit the remaining proof.
(3)⇐⇒(4) is obvious.
(3)=⇒(5) Let F ∈ M
∗
. Then F =

n∈N
F
n
where F
n
∈ N
∗
. Suppose that iF , ∅. Pick x ∈ iF. Then x ∈ U ⊂ F
for some U ∈ τ. Since F ∈ M
∗
, by Proposition 4.5, U ∈ M
∗
. By (3), U is ∗-second category in X. This is a
contradiction.

Frequently asked questions

Q1. What are the contributions in "On i-baire spaces" ?

In this paper, the concept of I-Baire spaces is introduced, and characterizations and properties of these spaces are given. 

Q2. What is the definition of a Baire space?

An operator (·)∗ : 2X −→ 2X, called a local function [17] of A with respect to τ and I, is defined as follows: for any A ⊂ X,A∗(I, τ) = {x ∈ X : U ∩ A < The authorfor every U ∈ τ(x)}2010 Mathematics Subject Classification. 

Q3. What is the meaning of the passage?

Since A ∈ N ∗(X), by Proposition 3.6, X−cA is ∗-dense in X. By Proposition 3.5, (X−cA)∩W , ∅. Note that (Y − cYA) ∩W = (Y − cA ∩ Y) ∩W = (Y − cA) ∩W = ((X − cA) ∩ Y) ∩ (U ∩ Y) = (X − cA) ∩W. By Proposition 3.5, Y − cYA is ∗-dense in Y. By Proposition 3.6, A ∈ N ∗(Y). 

Q4. What are some of the common applications of Baire spaces?

To develop applications of Baire spaces, some researchers have studied some spaces such as hyperspaces, Volterra spaces (see [10, 19]). 

Q5. What is the simplest way to prove that X is I-Baire?

Let {Gn} be a sequence of open and ∗-dense subsets of X. Put G = ∩n∈N Gn. For any n ∈ N andα ∈ Γ, the authors denote Gnα = Gn ∩ Xα and Gα = ∩ n∈N Gnα. 

Q6. What is the proof of the Lemma 6.11?

Then X is M-resolvable if and only if X has two disjoint dense I-Baire subspaces (i.e., X = A ∪ B, where A ∩ B = ∅, cA = cB = X, A and B are respectively M∗A-Baire and M∗B-Baire). 

Q7. What is the definition of a mapping?

A mapping f : (X, τ)→ (Y, σ) is called feebly open, if for any U ∈ τ − {∅}, i f (U) , ∅.Definition 7.5. ([13]) A mapping f : (X, τ,I) → (Y, σ) is called semi-I-continuous, if f−1(V) ∈ SIO(X, τ) for any V ∈ σ. 

Q8. What is the simplest way to explain the f-dense subsets?

{Un} is a sequence of open and ∗-dense subsets of X. Since (X, τ,I) is I-Baire, the authors obtain that c( ∩n∈N Un) = X. Note that f is continuous. 

Q9. what is the topological sum of x?

It is easy to prove that τ is a topology on X and Xα is clopen in X for any α ∈ Γ, and hence each Xα is ∗-closed and ∗-open in X.By Lemma 7.11, (X, τ,I) is an ideal space, which is called the topological sum of {(Xα, τα,Iα) : α ∈ Γ}. 

Q10. What is the simplest way to denote a subset of an ideal space?

Remark 3.2. Let (X, τ,I) be an ideal space and let B ⊂ A ⊂ X. (1) If A is ∗-dense in X, then A is dense in X. (2) If A is nowhere ∗-dense in X, then A is nowhere dense in X. (3) If A ∈ N ∗, then B ∈ N ∗. (4) A ∈ N ∗ if and only if cA ∈ N ∗.Proposition 3.3. 

Q11. what is the simplest definition of a Baire space?

Recall that a set in a Baire space is said to be a first category set or a meager set, if it can be written as a countable union of nowhere dense sets. 

Q12. What is the definition of a perfect space?

Then f (I) = { f (A) : A ∈ I} is an ideal on Y.Definition 7.2. ([13]) A subset A of an ideal space (X, τ,I) is called semi-I-open, if A ⊂ c∗iA. 

Q13. What is the definition of the -dense subset of an ideal space?

The family of all nowhere ∗-dense subsets of an ideal space X shall be denoted by N ∗(X) or simply by N ∗ when no ambiguity is present. 

Q14. what is the definition of a space?

Definition 4.1. Let (X, τ,I) be an ideal space and let A ⊂ X. (1) A is called ∗-first category or ∗-meager in X, if there exists a sequence {An} consisting of nowhere ∗-densesubsets of X such that A = ∪n∈N An.(2) A is called ∗-second category or ∗-nonmeager in X, if A is not ∗-first category in X. (3) A is called ∗-residual or ∗-comeager in X, if X − A is ∗-first category in X. 

Q15. What is the simplest way to prove that X is not I-Baire?

Since (X, τ,I) is I-Baire, by Theorem 5.5, U is ∗-second category in X. By Corollary 4.4, U is ∗-second category in Y. By Theorem 5.5, (Y, τY,IY) is IY-Baire.