1. Introduction
The notion of α-open set was introduced by Njastad in [2] . Caldas in [1] introduced and studied topological properties of α-derived, α-border, α-frontier, α-exterior of a set by using the concept of α-open sets. In this paper, we introduce and study the same above concepts by using ii-open sets. A subset A of X is called ii-open set [3] if there exists an open set G in the topology
of X, such that:
,
and
, the complement of an ii-open set is an ii-closed set. We denote the family of ii-open sets in
by
. It is shown in [4] that each of
and
is a topology on X. This property allows us to prove similar properties of α-open set. Also, we define ii-continuous functions and we study the relation between this type of function and continuous, semi-continuous, α-continuous and i-continuous functions. Finally, we introduce a new type of separation axioms namely
,
. We prove similar properties and characterizations of
and
.
2. Preliminaries
Throughout this paper,
and
(simply X and Y) always mean topological spaces. For a subset A of a space X, Cl (A) and Int (A) denote the closure of A and the interior of A respectively. We recall the following definitions, which are useful in the sequel.
Definition 2.1. A subset A of a space X is called
1) Semi-open set [5] if
.
2) α-open set [2] if
.
3) i-open set [3] if there exist an open set G in the topology
of X, such that
i)
ii)
The complement of an i-open set is an i-closed set.
4) ii-open set [4] if there exist an open set G in the topology
of X, such that
i)
ii)
iii)
The complement of an ii-open set is an ii-closed set.
5) int-open set [4] if there exist an open set G in the topology
of X and
such that
. The complement of int-open set is int-closed set.
6) αo (X), So (X), io (X), iio (X), into (X) are family of α-open, semi-open, i-open, ii-open, int-open sets respectively.
7)
,
denote the family of all i-open sets and ii-open sets respectively.
Definition 2.2. [3] A topological space X is called
1)
if a, b are to distinct points in X, there exist an i-open set U such that either
and
, or
and
.
2)
if
and
, there exist i-open sets U, V containing a, b respectively, such that
and
.
Definition 2.3. A function
is called
1) Continuous [6] , if
is open in
for every open set G of
.
2) α-continuous [6] , if
is α-open in
for every open set G of
.
3) Semi-Continuous [5] , if
is semi-open in
for every open set G of
.
4) i-Continuous [3] , if
is i-open in
for every open set G of
.
3. Applications of ii-Open Sets
Definition 3.1. Let A be a subset of a topological space
. A derived set of A denoted by D(A) is defined as follows:
. A point
is said to be ii-limit point of A if it satisfies the following assertion:
. The set of all ii-limit points of A is called the ii-derived set of A and is denoted by
. Note that
is not ii-limit point of A if and only if there exist an ii-open set G in X such that (
and
).
Theorem 3.2. For subsets A, B of a space X, the following statements hold:
1)
2) If
, then
3)
and
4)
5)
Proof. 1) Since every open set is ii-open [4] , it follows that
.
2) Let
. Then G is ii-open set containing x such that
(3.1)
Since
we get
, it implies that
, from (3.1) we get
.
Hence,
. Therefore
.
3) Since
and
, from (2) we get
,
.
This implies to
.
We shall prove that
. Since
,
, from (2) we get
and
. Therefore
.
4) If
and G is an ii-open set containing x, then
. Let
. Then, since
and
,
. Let
. Then,
for
and
. Hence,
. Therefore,
.
5) Let
. If
, the result is obvious. So, let,
then for ii-open set G containing x,
. Thus,
or
.
Now, it follows similarly from (4) that
. Hence,
.
Therefore, in any case,
.
In general, the converse of (1) may not true and the equality does not hold in (3) of theorem 3.2.
Example 3.3. Let
and
. Thus,
. Take the following:
1)
. Then,
and
. Hence,
;
2)
and
. Then
and
. Hence
.
Theorem 3.4. For any subset A of a space X,
.
Proof. Since
,
. On the other hand, Let
. If
, then the proof is complete. If
, each ii-open set G containing x intersects A at a point distinct from x; so
. Thus,
, which completes the proof.
Definition 3.5. A point
is said to be ii-interior point of A if there exist an ii-open set G containing x such that
. The set of all ii-interior points of A is said to be ii-interior of A and denoted by
.
Theorem 3.6. For subset A, B of a space X, the following statements are true:
1)
is the union of all ii-open subset of A
2) A is ii-open if and only if
3)
4)
5)
6)
7) If
, then
8)
9)
Proof. 1) Let
be a collection of all ii-open subsets of A. If
, then there exist
such that
. Hence
, and so
. On the other hand, if
, then
for some
. Thus
, and
. Accordingly,
.
2) Straightforward.
3) It follows from (1) and (2).
4) If
, then
and so there exist an ii-open set G containing x such that
. Thus,
and hence
. This shows that
. Now let
. Since
and
. We have
. Therefore,
.
5) Using (4) and Theorem (3.4), we have
.
6) Using (4) and Theorem (3.4), we get.
7) Since
and
,
, we get
.
8) Since
and
, from (7) we get
,
. Therefore
.
9) Since
and
, from (7) we get
,
. Therefore
.
Definition 3.7.
is said to be the ii-border of A.
Theorem 3.8. For a subset A of a space X, the following statements hold:
1)
where
denotes the border of A
2)
3)
4)
if and only if A is ii-open set
5)
6)
7)
8)
9)
Proof.
1) Since
, we have
.
2) and (3). Straightforward.
4) Since
, it follows from Theorem 3.6 (2). That A is ii-open
.
5) Since
is ii-open, it follows from (4) that
.
6) If
, then
. On the other hand, since
,
. Hence,
.
Which contradicts (3). Thus
.
7) Using (6), we get
.
8) Using Theorem 3.6 (6), we have
9) Applying (8) and the Theorem (3.4), we have
.
Example 3.9. Consider the topological space
given in Example (3.3). If
, then
and
. Hence,
, that is, in general, the converse Theorem 3.9 (1) may not be true.
Definition 3.10.
is said to be the ii-frontier of A.
Theorem 3.11. For a subset A of a space X, the following statements hold:
1)
where
denotes the frontier of A
2)
3)
4)
5)
6)
if and only if A is ii-open set
7)
8)
9)
is ii-closed
10)
11)
12)
13)
Proof.
1) Since
and
, it follows that
.
2)
.
3)
.
4) Since
, we have
.
5) Since
,
.
6) Assume that A is ii-open. Then
, by using (5), Theorem 3.6 (2), Theorem 3.8 (4) and Theorem 3.8 (9).
Conversely, suppose that
. Then
. by using (4) and (5) of Theorem 3.6, and so
. Since
in general, it follows that
so from Theorem 3.6 (2) that A is ii-open set.
7)
.
8) It follows from (7).
9)
Hence,
is ii-closed.
10)
.
11) Using Theorem 3.6 (3), we get
.
12)
13)
.
The converses of (1) and (4) of Theorem 3.11 are not true in general, as shown by Example
Example 3.12. Consider the topological space
given in Example 3.3. If
, then
, and if
, then
.
Definition 3.13.
is said to be an ii-exterior of A.
Theorem 3.14. For a subset A of a space X, the following statements hold:
1)
where
denotes the exterior of A
2)
is ii-open
3)
4)
5) If
, then
6)
7)
8)
9)
10)
11)
12)
Proof. 1) It follows from Theorem 3.6 (1).
2) It is straightforward by Theorem 3.6 (6).
3)
.
4) Assume that
. Then
, by using Theorem 3.6 (7).
5) Applying Theorem 3.6 (8), we get
6) Applying Theorem 3.6 (9), we obtain
.
7) Straightforward.
8) Straightforward.
9)
10)
Example 3.15. Let
and
. Thus,
. If
and
. Then
.
and
.
4. A New Decomposition of ii-Continuity
We begin by the following definition:
Definition 4.1. A function
is called ii-continuous if
is ii-open set in
for any open set G of
.
Theorem 4.2. Let
be a function then:
1) Every continuous function is an ii-continuous,
2) Every ii-continuous function is an i-continuous,
3) Every α-continuous function is an ii-continuous.
Proof. 1) Let G be open set in
. Since f is continuous, it follows that
is open set in
. But every open set is ii-open set [4] . Hence
is ii-open set in
. Thus f is ii-continuous.
2) Let G be open set in
. Since f is an ii-continuous, it follows that
is an ii-open set in
. But every ii-open set is i-open set [4] . Hence
is i-open set in
. Thus f is i-continuous.
3) Let G be open set in (
ϭ). Since f is α-continuous, it follows that
is α-open set in
. But every α-open set is ii-open set [4] . Hence
is ii-open set in
. Thus f is an ii-continuous.
The converse need not be true by the following example.
Example 4.3. Let
,
and
,
and
,
.
Let
be the identity function then
,
,
,
. Then f is ii-continuous, but f is not α-continuous, since for the open set
in
,
is not α-open in
and f is not continuous, since for the open set
in
,
is not open in
. Now when
be defined by
,
,
,
we get f is i-continuous, but f is not ii-continuous, since for the open set
in
,
is not ii-open in
.
Theorem 4.4. Let
be a function then every semi-continuous function is an ii-continuous.
Proof. Let G be open set in
. Since f is semi-continuous, it follows that
is semi-open set in
. But every semi-open set is ii-open set [4] . Hence
is ii-open set in
. Thus f is an ii-continuous.
Definition 4.5. A function
is called int-continuous if
is int-open set in
for any open set G in
.
Theorem 4.6. Let
be a function then:
1) Every continuous function is int-continuous,
2) Every ii-continuous function is int-continuous,
3) Every α-continuous function is int-continuous.
Proof. 1) Let G be open set in
. Since f is continuous, it follows that
is open set in
. But every open set is int-open set [4] . Hence
is int-open set in
. Thus f is int-continuous.
2) Let G be open set in
. Since f is ii-continuous, it follows that
is an ii-open set in
. But every ii-open set is int-open set [4] . Hence
is int-open set in
. Thus f is int-continuous.
3) Let G be open set in
. Since f is α-continuous, it follows that
is α-open set in
. But every α-open set is int-open set [4] . Hence
is int-open set in
. Thus f is int-continuous.
The converse need not be true by the following example.
Example 4.7. Let
,
and
,
and
,
. Let
be the identity function then
,
,
. Then f is int-continuous, but f is not ii-continuous, since for the open set
in
,
is not ii-open in
and f is not continuous, since for the open set
in
,
is not open in
and f is not α-continuous, since for the open set
in
,
is not α-open.
Definition 4.8. A subset A of X is called weakly ii-open set if A is ii-open set and
.
Theorem 4.9. A subset A of a space X is α-open set if and only if A is weakly ii-open.
Proof. Let A be α-open set. Since
and
. Therefore
, this implies that
. Now, put
where
, then A is ii-open set. Therefore, A is weakly ii-open set.
Conversely, Let A be weakly ii-open set, then there exist an open set
, such that
satisfying
and A is ii-open set. Since
, this implies that
and
. Since A is ii-open set, using (2) from Theorem (3.6), we get
. Therefore
. Thus A is α -open set.
As a summary the following Figure 1 shows the relations among semi-continuous, ii-continuous, i-continuous, int-continuous, α-continuous and continuous.
Figure 1. Relations among semi-continuous, ii-continuous, i-continuous, int-continuous, α-continuous and continuous.
Corollary 4.10. A function
is α-continuous if and only if it is weakly ii-continuous.
Proof. Clear from Theorem 4.9.
5. ii-Separating Axioms
In this section we define
and
spaces for ii-open sets and we determine them by giving many examples. Specially, we define
,
and
spaces to compare them with
space.
Definition 5.1. A topological space X is called
1)
if a, b are to distinct points in X, there exists ii-open set U such that either
and
, and
and
.
2)
if
and
, there exist ii-open sets U, V containing a, b respectively, such that
and
.
Example 5.2. Let
,
and
are topological spaces.
1)
(
) there exists
such that
,
. Therefore
is
.
2)
(
) there exists
such that
,
. Therefore
is
.
Theorem 5.3.
1) Every
-space is
-space,
2) Every
-space is
-space,
3) Every
-space is
-space,
4) Every
-space is
-space.
Proof. (1), (2), (3) and (4) follow using the fact that every open set is ii-open [4] .
The converse needs not to be true by the following example.
Example 5.4. Let
,
and
.
and
are topological spaces.
is not
-space because,
(
) there is no open set G such that
,
.
is
-space because,
(
) there exists
such that
,
.
(
) there exists
such that
,
.
(
) there exists
such that
,
.
is not
-space because,
(
) there exists
such that
,
.
is not
-space because,
(
) there exists
such that
,
.
Theorem 5.5. Every
-space is
-space.
Proof. Let X be
-space. Let a, b be two distinct points in X. Since X is
-space there exist two α-open sets U, V in X such that
,
,
,
. Since every α-open set is ii-open set [4] , U, V is an ii-open set in X. Hence X is
-space.
Theorem 5.6. Every
-space is
-space.
Proof. Let X be a
-space. Let a, b be two distinct points in X. Since X is
-space there exist two ii-open sets U, V in X such that
,
,
,
. Since every ii-open set is i-open set [4] , U, V is an i-open set in X. Hence X is
-space.
The converse needed not to be true by the following example.
Example 5.7. Let
,
and.
.
.
and
are topological spaces.
is
-space because,
(
) there exists
such that
,
and
,
.
(
) there exists
such that
,
and
,
.
(
) there exists
such that
,
and
,
.
is not
-space because,
(
) there exists
such that
,
.
Theorem 5.8. A space X is
if and only if
for every pair of distinct points x, y of X.
Proof. Let X be a
-space. Let
such that
, then there exists an ii-open set U containing one of the points but not the other, then
and
. Then
is ii-closed set containing y but not x. But
is the smallest ii-closed set containing y. Therefore
and hence
. Thus
.
Conversely, Suppose for any
with
,
. Let
such that
but
. If
then
and hence
. This is contradiction. Therefore
. That is
. Therefore
is ii-open set containing x but not y. Hence X is an
-space.
Theorem 5.9. A space
is
-space if and only if the singletons are ii-closed sets.
Proof. Let X be
-space and let
, to prove that
is ii-closed set. We will prove
is ii-open set in X. Let
, implies
and since X is
-space then their exist two ii-open sets U, V such that
,
. Since
, then
is ii-open set. Hence
is ii-closed set.
Conversely, Let
then
are ii-closed sets. That is
is ii-open set clearly,
and
. Similarly
is ii-open set,
and
. Hence X is an
-space.
As a consequence the following Figure 2 shows the relations among
,
,
,
and
.
Figure 2. Relations among
,
,
,
and
.