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Abstract
In this paper, a new kind of sets called Dα-open sets are introduced and studied in a topological spaces. The class of all Dα-open sets is strictly between the class of all α-open sets and g-open sets. Also, as applications we introduce and study Dα-continuous, Dα-open, and Dα-closed functions between topological spaces. Finally, some properties of Dα-closed and strongly Dα-closed graphs are investigated.
1 Introduction and preliminaries
Generalized open sets play a very important role in General Topology, and they are now the research topics of many topologies worldwide. Indeed a significant theme in General Topology and Real Analysis is the study of variously modified forms of continuity, separation axioms, etc. by utilizing generalized open sets. One of the most well-known notions and also inspiration source are the notion of α-open [Citation1] sets introduced by Njåstad in 1965 and generalized closed (g-closed) subset of a topological space [Citation2] introduced by Levine in 1970. Since then, many mathematicians turned their attention to the generalization of various concepts in General Topology by considering α-open sets [Citation3–Citation[4]Citation[5]Citation[6]Citation[7]Citation[8]Citation[9]Citation10] and generalized closed sets [Citation11–Citation[12]Citation13]. In 1982 Dunham [Citation14] used the generalized closed sets to define a new closure operator, and thus a new topology τ*, on the space, and examined some of the properties of this new topology. Throughout the present paper and (Z, ν) denote topological spaces (briefly X, Y and Z) and no separation axioms are assumed on the spaces unless explicitly stated. For a subset A of a space (X, τ), Cl(A) and Int(A) denote the closure and the interior of A, respectively. Since we require the following known definitions, notations, and some properties, we recall in this section.
Definition 1.1
Let (X, τ) be a topological space and A ⊆ X. Then
(i) |
(ii) | A is generalized closed (briefly g-closed) [Citation2] if Cl(A) ⊆ U whenever A ⊆ U and U is open in X. |
(iii) | A is generalized open(briefly g-open) [Citation2] if X\A is g-closed. |
The α-closure of a subset A of X [Citation3] is the intersection of all α-closed sets containing A and is denoted by Clα(A). The α-interior of a subset A of X [Citation3] is the union of all α-open sets contained in A and is denoted by Intα(A). The intersection of all g-closed sets containing A [Citation14] is called the g-closure of A and denoted by Cl*(A), and the g-interior of A [Citation15] is the union of all g-open sets contained in A and is denoted by Int*(A).
We need the following notations:
• | αO(X) (resp. αC(X)) denotes the family of all α-open sets (resp. α-closed sets) in (X, τ). |
• | GO(X) (resp. GC(X)) denotes the family of all generalized open sets (resp. generalized closed sets) in (X, τ). |
• |
|
Definition 1.2
A function f : X → Y is said to be:
(i) | α-continuous [Citation16] (resp. g-continuous [Citation17]) if the inverse image of each open set in Y is α-open (resp. g-open) in X. |
(ii) | α-open [Citation16] (resp. α-closed [Citation16]) if the image of each open (resp. closed ) set in X is α-open (resp. α-closed) in Y. |
(iii) | g-open [Citation18] (resp. g-closed [Citation18]) if the image of each open (resp. closed) set in X is g-open (resp. g-closed) in Y. |
Definition 1.3
Let f : X → Y be a function:
(i) | The subset |
(ii) | a closed graph [Citation19] if its graph G(f) is closed sets in the product space X × Y. |
(iii) | a strongly closed graph [Citation20] if for each point |
(iv) | a strongly α-closed graph [Citation21] if for each |
Definition 1.4
A topological space (X, τ) is said to be:
(i) | α-T1 [Citation9] (resp. g-T1 [Citation22]) if for any distinct pair of points x and y in X, there exist α-open (resp. g-open) set U in X containing x but not y and an α-open (resp. g-open) set V in X containing y but not x. |
(ii) | α-T2 [Citation8] (resp. g-T2 [Citation22]) if for any distinct pair of points x and y in X, there exist α-open (resp. g-open) sets U and V in X containing x and y, respectively, such that U ∩ V = ϕ. |
Lemma 1.5
Let A ⊆ X, then
(i) |
|
(ii) |
|
Lemma 1.6
A function has a closed graph [Citation19] if for each
, there exist U ∈ O(X, x) and V ∈ O(Y, y) such that f(U) ∩ V = ϕ.
Lemma 1.7
The graph G(f) is strongly closed [Citation23] if and only if for each point , there exist open sets U ⊂ X and V ⊂ Y containing x and y, respectively, such that
.
2 Dα-closed sets
In this section we introduce Dα-closed sets and investigate some of their basic properties.
Definition 2.1
A subset A of a space X is called Dα-closed if .
The collection of all Dα-closed sets in X is denoted by DαC(X).
Lemma 2.2
If there exists an g-closed set F such that , then A is Dα-closed.
Proof
Since F is g-closed, Cl*(F) = F. Therefore, . Hence A is Dα-closed.
Remark 2.3
The converse of above lemma is not true as shown in the following example.
Example 2.4
Let (X, τ) be a topological space, where X = {a, b, c} and . Then
,
,
,
,
. Therefore
and
but
.
Theorem 2.5
Let (X, τ) be a topological space. Then
(i) | Every α-closed subset of (X, τ) is Dα-closed. |
(ii) | Every g-closed subset of (X, τ) is Dα-closed. |
Proof
(i) Since closed set is g-closed, [Citation14]. Now, suppose A is α-closed in X, then
. Therefore,
. Hence A is Dα-closed in X.
(ii) Suppose A is g-closed. Then Cl*(A) = A [Citation14]. Therefore, . Then
[Citation14]. Hence A is Dα-closed.
Remark 2.6
The converse of above theorem is not true as shown in the following example.
(i) | Dα-closed set need not be α-closed. (see Example 2.7 below) |
(ii) | Dα-closed set need not be g-closed. (see Example 2.7 below) |
Example 2.7
Let (X, τ) be a topological space, where and
. Then
,
,
,
,
,
. Therefore
, but
and
.
From the above discussions we have the following diagram in which the converses of implications need not be true.
Theorem 2.8
Arbitrary intersection of Dα-closed sets is Dα-closed.
Proof
Let {Fi : i ∈ Λ} be a collection of Dα-closed sets in X. Then for each i. Since ∩ Fi ⊆ Fi for each i,
for each i. Hence
. Therefore
. Hence ∩Fi is Dα-closed.
Remark 2.9
The union of two Dα-closed sets need not to be Dα-closed as shown in Example 2.7, where both {a} and {b} are Dα-closed sets but is not Dα-closed.
Corollary 2.10
If a subset A is Dα-closed and B is α-closed, then A ∩ B is Dα-closed.
Proof
Follows from Theorem 2.5 (i) and Theorem 2.8.
Corollary 2.11
If a subset A is Dα-closed and F is g-closed, then A ∩ F is Dα-closed.
Proof
Follows from Theorem 2.5 (ii) and Theorem 2.8.
Definition 2.12
Let A be a subset of a space X. The Dα-closure of A, denoted by , is the intersection of all Dα-closed sets in X containing A. That is
.
Theorem 2.13
Let A be a subset of X. Then A is Dα-closed set in X if and only if .
Proof
Suppose A is Dα-closed set in X. By Definition 2.12, . Conversely, suppose
. By Theorem 2.8 A is Dα-closed.
Theorem 2.14
Let A and B be subsets of X. Then the following results hold.
(i) |
|
(ii) |
|
(iii) | If A ⊆ B, Then |
(iv) |
|
(v) |
|
(vi) |
|
Proof
(i) Follows From Theorem 2.5 (i) and (ii), respectively.
(ii) and (iii) are obvious.
(iv) If A ⊆ F, F ∈ DαC(X), then from (iii) and Theorem 2.13, . Again
. Therefore
.
(v) and (vi) follows from (iii).
Remark 2.15
The equality in the statements (v) of the above theorem need not be true as seen from Example 2.7, where A = {a}, B = {b}, and . Then one can have that,
;
;
;
. Further more the equality in the statements (iv) of the above theorem need not be true as shown in the following example.
Example 2.16
Let (X, τ) be a topological space, where X = {a, b, c} and . Then
,
. Let A = {a}, B = {b}, and A ∩ B = ϕ. Then one can have that,
;
;
;
.
3 Dα-open sets
In this section we introduce Dα-open sets and investigate some of their basic properties.
Definition 3.1
A subset A of a space X is called an Dα-open if X \ A is Dα-closed. Let DαO(X) denote the collection of all an Dα-open sets in X.
Lemma 3.2
Let A ⊆ X, then
(i) |
|
(ii) |
|
Proof
Obvious.
Theorem 3.3
A subset A of a space X is Dα-open if and only if .
Proof
Let A be Dα-open set. Then X \ A is Dα-closed and . By Lemma 3.2
. Conversely, suppose
. Then
. Hence
. This shows that X \ A is Dα-closed. Thus A is Dα-open.
Lemma 3.4
If there exists g-open set V such that , then A is Dα-open.
Proof
Since V is g-open, X \ V is g-closed and . Therefore From Lemma 3.2
. From Lemma 2.2 we have X \ A is Dα-closed. Hence A is Dα-open.
Remark 3.5
The converse of Lemma 3.4 need not to be true as seen from Example 2.4, where and
but
.
Theorem 3.6
Let (X, τ) be a topological space. Then
(i) | Every α-open subset of (X, τ) is Dα-open. |
(ii) | Every g-open subset of (X, τ) is Dα-open. |
Proof
From Theorem 2.5, the proof is obvious.
Remark 3.7
The converse of the above theorem is not true as seen from Example 2.7, where but
and
.
From the above discussions we have the following diagram in which the converses of implications need not be true.
Theorem 3.8
Arbitrary union of Dα-open set is Dα-open.
Proof
Follows from Theorem 2.8.
Remark 3.9
The intersection of two Dα-open sets need not be Dα-open as seen from Example 2.7, where both {b, c} and {a, c} are Dα-open sets but is not Dα-open.
Corollary 3.10
If a subset A is Dα-open and B is α-open, then A ∪ B is Dα-open.
Proof
Follows from Theorem 3.6 (i) and Theorem 3.8.
Corollary 3.11
If a subset A is Dα-open and U is g-open, then A ∪ U is Dα-open.
Proof
Follows from Theorem 3.6 (ii) and Theorem 3.8.
Definition 3.12
Let A be a subset of a space X. The Dα-interior of A is denoted by , is the union of all an Dα-open sets in X contained in A. That is
.
Lemma 3.13
If A is a subset of X, then
(i) |
|
(ii) |
|
Proof
Obvious.
Theorem 3.14
Let A be a subset of X. Then A is Dα-open if and only if .
Proof
Follows from Theorem 2.13 and Lemma 3.13.
Theorem 3.15
Let A and B be subsets of X. Then the following results hold.
(i) |
|
(ii) |
|
(iii) | If A ⊆ B, then |
(iv) |
|
(v) |
|
(vi) |
|
Proof
Obvious.
Remark 3.16
The equality in the statements (v) of Theorem 3.15 need not be true as seen from Example 2.7, where A = {b, c}, B = {a, c}, and A ∪ B = X. Then one can have that, ;
;
;
. Furthermore the equality in the statements (iv) of the above theorem need not be true as seen from Example 2.7, where A = {b, c}, B = {a, c}, and A ∩ B = {c}. Then one can have that,
;
;
;
.
Theorem 3.17
Let x ∈ X. Then if and only if U ∩ A ≠ ϕ for every Dα-open set U containing x.
Proof
Let and there exists Dα-open set U containing x such that U ∩ A = ϕ. Then
and X\U is Dα-closed. Therefore
. This implies
, which is a contradiction. Conversely, assume that U ∩ A ≠ ϕ for every Dα-open set U containing x and
. Then there exists Dα-closed subset F containing A such that
. Hence x ∈ X \ F and X \ F is Dα-open. Therefore A ⊆ F,
This is a contradiction to our assumption.
Lemma 3.18
Let A be any subset of (X, τ). Then
(i) |
|
(ii) |
|
Proof
(i) |
|
(ii) | From (i) we have |
Theorem 3.19
If A is a subset of a topological space X,
(i) |
|
(ii) |
|
Proof
(i) | Let |
(ii) | By Lemma 3.13 we have |
.
4 Dα-continuous functions
In this section we introduce Dα-continuous functions and investigate some of their basic properties.
Definition 4.1
A function f : X → Y is called Dα-continuous if the inverse image of each open set in Y is Dα-open in X.
Theorem 4.2
(i) | Every α-continuous function is Dα-continuous. |
(ii) | Every g-continuous function is Dα-continuous. |
Proof
It is obvious from Theorem 3.6.
Remark 4.3
(i) | Dα-continuous function need not be α-continuous. |
(see Example 4.4 (i) below)
(ii) | Dα-continuous function need not be g-continuous. |
(see Example 4.4 (ii) below)
Example 4.4
(i) Let X = {a, b, c} associated with the topology and Y = {x, y, z} associated with the topology
. Let f : X → Y be a function defined by f(a) = f(b) = x, f(c) = z. One can have that
,
,
,
, DαC(X) = DαO(X) =P(X). Since {z} is open in Y,
, but
. Therefore f is Dα-continuous but not α-continuous.
(ii) Let (X,τ) and (Y,σ) be the topological spaces in (i) and f : X → Y be a function defined by f(a) = x, f(b) = f(c) = z. Since {z} is open in Y, , but
. Therefore f is Dα-continuous but not g-continuous.
From the above discussions we have the following diagram in which the converses of implications need not be true.
Theorem 4.5
Let f : X → Y be a function. Then the following are equivalent:
(i) | f is Dα-continuous. |
(ii) | For each x ∈ X and each open set V ⊂ Y containing f(x), there exists Dα-open set W ⊂ X containing x such that f(W) ⊂ V. |
(iii) | The inverse image of each closed set in Y is Dα-closed in X. |
(iv) |
|
(v) |
|
(vi) |
|
Proof
(i)⇒(ii) Since V ⊂ Y containing f(x) is open, then . Set
which contains x, therefore f(W) ⊂ V.
(ii)⇒(i) Let V ⊂ Y be open, and let , then f(x) ∈ V and thus there exists
such that x ∈ Wx and f(Wx) ⊂ V. Then
, and so
but
by Theorem 3.8. Hence
, and therefore f is Dα-continuous.
(i)⇒(iii) Let F ⊂ Y be closed. Then Y\F is open and , i.e.
. Then
is Dα-closed of X.
(iii)⇒(iv) Let and F be a closed set in Y containing f(A). Then by (iii),
is Dα-closed set containing A. It follows that
and hence
. Therefore
.
(iv)⇒(v) Let and
. Then by assumption,
. This implies that
. Hence
.
(v)⇒(vi) Let . By assumption,
. This implies that,
and hence
. By taking complement on both sides we get
.
(vi)⇒(i) Let U be any open set in Y. Then Int(U) = U. By assumption, and hence
. Then
. Therefore by Theorem 3.14,
is Dα-open in X. Thus f is Dα-continuous.
Theorem 4.6
Let f : X → Y be Dα-continuous and let g : Y → Z be continuous. Then gof : X → Z is Dα-continuous.
Proof
Obvious.
Remark 4.7
Composition of two Dα-continuous functions need not be Dα-continuous as seen from the following example.
Example 4.8
Let associated with the topology
,
associated with the topology
and
associated with the topology
and
by f(a) = y, f(b) = x, f(c) = z. Define
by
, g(z) = r. One can have that
,
,
,
, and
,
,
. Clearly, f and g are Dα-continuous. {r} is open in Z. But
, which is not Dα-open in X. Therefore gof is not Dα-continuous.
5 Dα-open functions and Dα-closed functions
In this section we introduce Dα-open functions and Dα-closed functions and investigate some of their basic properties.
Definition 5.1
A function f : X → Y is said to be Dα-open (resp. Dα-closed) if the image of each open (resp. closed) set in X is Dα-open (resp. Dα-closed) in Y.
Theorem 5.2
(i) | Every α-open function is Dα-open. |
(ii) | Every g-open function is Dα-open. |
Proof
It is obvious from Theorem 3.6.
Remark 5.3
(i) | Dα-open function need not be α-open. (see Example 5.4 below) |
(ii) | Dα-open function set need not be g-open. (see Example 5.5 below) |
Example 5.4
(i) Let X = {x,y,z} associated with the topology and Y = {a,b,c} associated with the topology
. Let
be a function defined by f(x) = a, f(y) = b and f(z) = c. One can have that
, GC(Y) = GO(Y) = DαC(Y) = DαO(Y) = P(X). Since {x} is open in X,
, but
. Therefore f is Dα-open function but not α-open.
Example 5.5
(ii) Let X = {x,y,z} associated with the topology and Y = {a, b, c} associated with the topology
. Let
be a function defined by f(x) = b, f(y) = c and f(z) = a. One can have that
,
,
,
. Since {x,y} is open in X,
, but
. Therefore f is Dα-open function but not g-open.
From the above discussions we have the following diagram in which the converses of implications need not be true.
Theorem 5.6
Let f : X → Y be a function. The following statements are equivalent.
(i) | f is Dα-open. |
(ii) | For each x ∈ X and each neighborhood U of x, there exists Dα-open set W ⊆ Y containing f(x) such that W ⊆ f(U). |
Proof
(i)⇒(ii) Let x ∈ X and U is a neighborhood of x, then there exists an open set V ⊆ X such that x ∈ V ⊆ U. Set W = f(V). Since f is Dα-open, and so
.
(ii)⇒(i) Obvious.
Theorem 5.7
Let f : X → Y be Dα-open (resp. Dα-closed) function and W ⊆ Y. If A ⊆ X is a closed (resp. open) set containing , then there exists Dα-closed (resp. Dα-open) set H ⊆ Y containing W such that
.
Proof
Let . Since
, we have
. Since f is Dα-open (resp. Dα-closed), then H is Dα-closed (resp. Dα-open) set and
.
Corollary 5.8
If f : X → Y is Dα-open, then for each set B ⊂ Y.
Proof
Since is closed in X containing
for a set B ⊆ Y. By Theorem 5.7, there exists Dα-closed set H ⊆ Y, B ⊆ H such that
. Thus,
.
Theorem 5.9
A function f : X → Y is Dα-open if and only if for every subset A of X.
Proof
Suppose f : X → Y is Dα-open function and A ⊆ X. Then Int(A) is open set in X and is Dα-open set contained in f(A). Therefore
. Conversely, let be
for every subset A of X and U is open set in X. Then Int(U) = U,
. Hence
. By Theorem 3.14 f(U) is Dα-open.
Theorem 5.10
For any bijective function the following statements are equivalent.
(i) |
|
(ii) | f is Dα-open function. |
(iii) | f is Dα-closed function. |
Proof
(i)⇒(ii) Let U be an open set in X. Then X \ U is closed in X. Since is Dα-continuous,
is Dα-closed in Y. That is
is Dα-closed in Y. This implies f(U) is Dα-open in Y. Hence f is Dα-open function.
(ii)⇒ (iii) Let F be a closed set in X. Then X \ F is open in X. Since f is Dα-open, f(X \ F) is Dα-open in Y. That is is Dα-open in Y. This implies f(F) is Dα-closed in Y. Hence f is Dα-closed function.
(iii)⇒ (i) Let F be closed set in X. Since f is Dα-closed function, f(F) is Dα-closed in Y. That is is Dα-closed in Y. Hence
is Dα-continuous function.
Remark 5.11
Composition of two Dα-open functions need not be Dα-open as seen from the following example.
Example 5.12
Let X = {x, y, z} associated with the topology , Y = {p, q, r} associated with the topology
and Z = {a,b,c} associated with the topology,
. Define
by f(x) = p, f(y) = q, f(z) = r and
by g(p) = b, g(q) = a, g(r) = c. One can have that;
,
,
,
and
,
,
,
,
. Clearly, f and g are Dα-open function. {z} is open in X. But
which is not Dα-open in Z. Therefore gof is not Dα-open function.
6 Dα-closed graph and strongly Dα-closed
In this section we introduce Dα-closed graph and strongly Dα-closed and investigate some of their basic properties.
Definition 6.1
A function f : X → Y has Dα-closed graph if for each , there exist
and
such that
.
Remark 6.2
Evidently every closed graph is Dα-closed. That the converse is not true is seen from the following example.
Example 6.3
Let X = {a, b, c} associated with the topology and Y = {x, y, z} associated with the topology
. Let
be a function defined by f(a) = f(c) = x, f(b) = y. One can have that
,
,
,
and
,
,
. Since
and
but
and
. Therefore G(f) is Dα-closed but not closed.
Theorem 6.4
Let be a function and
(i) | f is Dα-closed graph; |
(ii) | For each |
(iii) | For each |
(iv) | For each
|
Proof
(i)⇒(ii) Suppose f is Dα-closed graph. Then for each , there exists
and
such that
. This implies that for each
and y ∈ Cl*(V). Since y ≠ f(x),
.
(ii)⇒(i) Let . Then there exists
and
such that
. This implies that f(x) ≠ y for each x ∈ U and y ∈ Cl*(V). Therefore
.
(i)⇒(iii) Suppose f is Dα-closed graph. Then for each , there exists
and
such that
. Since g-open set is Dα-open,
. Therefore
.
(ii)⇒(iv) Let . Then there exists
and
such that
. Since
,
.
(i)⇒(iv) From (ii).
Definition 6.5
A topological space (X,τ) is said to be Dα-T1 if for any distinct pair of points x and y in X, there exist Dα-open U in X containing x but not y and an Dα-open V in X containing y but not x.
Theorem 6.6
(i) | Every α-T1 space is Dα-T1. |
(ii) | Every g-T1 space is Dα-T1. |
Proof
It is obvious from Theorem 3.6.
Remark 6.7
The converse of the above theorem is not true as seen from Example 2.7.
Theorem 6.8
Let f : X → Y be any surjection with G(f) Dα-closed. Then Y is g-T1.
Proof
Let . The subjectivity of f gives the existence of an element xo ∈ X such that f(xo) = y2. Now
. The Dα-closeness of G(f) provides
,
such that
. Now
. This and the fact that
guarantee that
. Again from the subjectivity of f gives a x1 ∈ X such that f(x1) = y1. Now
and the Dα-closedness of G(f) provides
,
such that
. Now
so that
. Thus we obtain sets
such that y1 ∈ V1 but
while y2 ∈ V2 but
. Hence Y is g-T1.
Corollary 6.9
Let f : X → Y be any surjection with G(f) Dα-closed. Then Y is Dα-T1.
Proof
Follows From Theorems 6.6 (i) and 6.8.
Theorem 6.10
Let f : X → Y be any injective with G(f) Dα-closed. Then X is Dα-T1.
Proof
Let . The injectivity of f implies
whence one obtains that
. The Dα-closedness of G(f) provides
,
such that
. Therefore
and a fortiori
. Again
and Dα-closedness of G(f) as before gives
,
with
, which guarantees that
and so
. Therefore, we obtain sets U1 and
such that x1 ∈ U1 but
while x2 ∈ U2 but
. Hence X is Dα-T1.
Corollary 6.11
Let f : X → Y be any bijection with G(f) Dα-closed. Then both X and Y are Dα-T1.
Proof
It readily follows from Corollary 6.9 and Theorem 6.10.
Definition 6.12
A topological space (X, τ) is said to be Dα-T2 if for any distinct pair of points x and y in X, there exist Dα-open sets U and V in X containing x and y, respectively, such that U ∩ V = ϕ.
Theorem 6.13
(i) | Every α-T2 space is Dα-T2. |
(ii) | Every g-T2 space is Dα-T2. |
Proof
Obvious.
Remark 6.14
The converse of the above theorem is not true as seen from Example 2.7.
Theorem 6.15
Let f : X → Y be any surjection with G(f) Dα-closed. Then Y is g-T2.
Proof
Let . The subjectivity of f gives a x1 ∈ X such that f(x1) = y1. Now
. The Dα-closedness of G(f) provides
,
such that
. Now
. This and the fact that
guarantee that
. This mean that there exists
such that W ∩ V = ϕ. Hence Y is g-T2.
Corollary 6.16
Let f : X → Y be any surjection with G(f) Dα-closed. Then Y is Dα-T2.
Proof
Follows from Theorems 6.13 (ii) and 6.15.
Definition 6.17
A function f : X → Y has a strongly Dα-closed graph if for each , there exist
and V ∈ O(Y, y) such that
.
Corollary 6.18
A strongly Dα-closed graph is Dα-closed. That the converse is not true is seen from Example 6.3, where but
. Therefore G(f) is Dα-closed but not strongly Dα-closed.
Remark 6.19
Evidently every strongly α-closed graph (resp. strongly closed graph) is strongly Dα-closed graph. That the converse is not true is seen from the following example.
Example 6.20
Let X = {a, b, c} associated with the topology and Y = {x, y, z} associated with the topology
. Let
be a function defined by f(a) = f(c) = x, f(b) = y. One can have that
,
,
,
,
. Since
and
but
(resp.
). Therefore G(f) is strongly Dα-closed but not strongly α-closed (resp. strongly closed).
Theorem 6.21
For a function , the following properties are equivalent:
(i) | f has strongly Dα-closed graph. |
(ii) | For each |
(iii) | For each |
(iv) | For each |
Proof
Similar to the proof of Theorem 6.4.
Theorem 6.22
If f : X → Y is a function with a strongly Dα-closed graph, then for each x ∈ X, .
Proof
Suppose the theorem is false. Then there exists a y ≠ f(x) such that . This implies that
for every
. So V ∩ f(U) ≠ ϕ for every
. This, in its turn, indicates that
, which contradicts the hypothesis that f is a function with Dα-closed graph. Hence the theorem holds.
Theorem 6.23
If f : X → Y is Dα-continuous function and Y is T2. Then G(f) is strongly Dα-closed.
Proof
Let . Since Y is T2, there exists a set V ∈ O(Y, y) such that
. But Cl(V) is closed. Now
. By Theorem 4.5 there exists
such that
. Consequently,
and therefore G(f) is strongly Dα-closed.
Theorem 6.24
Let f : X → Y be any surjection with G(f) strongly Dα-closed. Then Y is T1 and α-T1.
Proof
Similar to the proof of Theorem 6.8 and T1-ness always guarantees α-T1-ness. Hence Y is α-T1.
Corollary 6.25
Let f : X → Y be any surjection with G(f) strongly Dα-closed. Then Y is Dα-T1.
Proof
Follows From Theorems 6.6 (i) and 6.24.
Theorem 6.26
Let f : X → Y be any injective with G(f) strongly Dα-closed. Then X is Dα-T1.
Proof
Similar to the proof of Theorem 6.10.
Corollary 6.27
Let f : X → Y be any bijection with G(f) strongly Dα-closed. Then both X and Y are Dα-T1.
Proof
It readily follows from Corollary 6.25 and Theorem 6.26.
Theorem 6.28
Let f : X → Y be any surjection with G(f) strongly Dα-closed. Then Y is T2 and α-T2.
Proof
Similar to the proof Theorem 6.15 and T2-ness always guarantees α-T2-ness. Hence Y is α-T2.
Corollary 6.29
Let f : X → Y be any surjection with G(f) strongly Dα-closed. Then Y is Dα-T2.
Proof
Follows From Theorems 6.13 (i) and 6.28.
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