Download On z-θ-Open Sets and Strongly θ-z

Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 8, 355 - 367
HIKARI Ltd, www.m-hikari.com
On z-θ-Open Sets and Strongly
θ-z-Continuous Functions
Murad Özkoç
Muğla Sıtkı Koçman University
Faculty of Science Department of Mathematics
48000 Muğla/TURKEY
[email protected]
c 2013 Murad Özkoç. This is an open access article distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
A new class of generalized open sets, called z-open sets which are
weaker than both θ-semiopen sets and θ-preopen sets defined by Altunöz and Aslım [2], is introduced and some properties are obtained by
Özkoç [14]. In this paper in order to investigate some different properties we introduce two strong form of z-open sets called z-regular sets
and z-θ-open sets. By means of z-θ-open sets we also introduce a new
class of functions called strongly θ-z-continuous functions which is a
generalization of strongly θ-precontinuous functions. We obtain some
characterizations and several properties of such functions. Moreover,
we define strongly z-closed graphs, z-regular spaces and z-T2 spaces
and investigate the relationship between their properties and strongly
θ-z-continuous functions.
Mathematics Subject Classification: Primary 54C08, 54C10
Keywords: z-open sets, z-θ-open sets, z-regular sets, strongly θ-z-continuous
functions
1
Introduction
The notion of continuity is an important concept in general topology. Some
properties of strongly θ-continuous functions defined via θ-open sets are studied by Long and Herrington [9]. Recently, five generalizations of strong θcontinuity are obtained by Jafari and Noiri [7], Noiri [12], Noiri and Popa [13],
356
Murad Özkoç
Park [16] and Özkoç [15]. The main goal of this paper is to introduce and
investigate some fundamental properties of strongly θ-z-continuous functions
defined via z-open sets introduced by Özkoç [14] in a topological space. It turns
out that strong θ-z-continuity is weaker than strong θ-precontinuity [12].
2
Preliminaries
Throughout the present paper, spaces X and Y always mean topological
spaces. Let X be a topological space and A a subset of X. The closure
of A and the interior of A are denoted by cl(A) and int(A), respectively.
All open neighborhoods of the point x of X is denoted by U(x). A subset
A is said to be regular open (resp. regular closed) if A = int(cl(A)) (resp.
A = cl(int(A))). The subset A is called θ-open [18] if A = intθ (A), where
intθ (A) = {x|(∃U ∈ U(x))(cl(U) ⊂ A)}. The complement of a θ-open set is
called θ-closed. The family of all θ-open (resp. θ-closed) sets in X is denoted
by θO(X) (resp. θC(X)).
The z-interior [14] of a subset A of X is the union of all z-open sets of X
contained in A and is denoted by z-int(A). The z-closure [14] of a subset A of
X is the intersection of all z-closed sets of X containing A and is denoted by
z-cl(A).
A subset A of X is called α-open [11] (resp., semiopen [8], θ-semiopen
[2], preopen [10], θ-preopen [2], b-open [4], e-open [6], z-open [14], semipreopen [3] (or β-open [1])) if A ⊂ int(cl(int(A))) (resp. A ⊂ cl(int(A)), A ⊂
cl(intθ (A)), A ⊂ int(cl(A)), A ⊂ int(clθ (A)), A ⊂ int(cl(A)) ∪ cl(int(A)), A ⊂
int(clδ (A)) ∪ cl(intδ (A)), A ⊂ int(clθ (A)) ∪ cl(intθ (A)), A ⊂ cl(int(cl(A))))
and the complement of an α-open (resp. semiopen, θ-semiopen, preopen, θpreopen, b-open, e-open, z-open, semi-preopen, (or β-open)) set are called
α-closed (resp. semiclosed, θ-semiclosed, preclosed, θ-preclosed, b-closed, eclosed, z-closed, semi-preclosed (or β-closed)).
The intersection of all semiclosed (resp. preclosed, θ-semiclosed, θ-preclosed,
b-closed, e-closed, z-closed) sets of X containing A is called the semi-closure
[8] (resp. pre-closure [10], θ-semi-closure [2], θ-pre-closure [2], b-closure [4],
e-closure [6], z-closure [14]) of A and is denoted by scl(A) (resp. pcl(A), θscl(A), θ-pcl(A), bcl(A), e-cl(A), z-cl(A)). Dually, the semi-interior (resp. preinterior, θ-semi-interior, θ-pre-interior, b-interior, e-interior, z-interior) of A is
defined to be the union of all semiopen (resp. preopen, θ-semiopen, θ-preopen,
b-open, e-open, z-open) sets contained in A and is denoted by sint(A) (resp.
pint(A), θ-sint(A), θ-pint(A), bint(A), e-int(A), z-int(A)). The family of all
θ-semiopen (resp. θ-preopen, b-open, e-open, z-open) sets in X is denoted by
θSO(X) (resp. θP O(X), BO(X), eO(X), zO(X)).
Lemma 2.1. ([2], [14]). Let A and B be any subsets of a space X. Then
On z-θ-open sets and strongly θ-z-continuous functions
357
the following hold:
(a) θ-scl(A) = A ∪ int(clθ (A)), θ-sint(A) = A ∩ cl(intθ (A)),
(b) θ-pcl(A) = A ∪ cl(intθ (A)), θ-pint(A) = A ∩ int(clθ (A)),
(c) z-cl(A) = θ-scl(A) ∩ θ-pcl(A).
3
z-regular Sets and z-θ-open Sets
Definition 3.1. A subset A of a topological space X is z-regular if it is both
z-open and z-closed. The family of all z-regular sets (resp. which involve the
point x of X) in X will be denoted by zR(X)(resp. zR(X, x)).
Theorem 3.2. The following properties hold for a subset A of a topological
space X.
(a) A ∈ zO(X) if and only if z-cl(A) ∈ zR(X).
(b) A ∈ zC(X) if and only if z-int(A) ∈ zR(X).
Proof. We will prove only the first statement. The second one can be proved
similarly.
Necessity. Let A ∈ zO(X). Then we have A ⊂ int(clθ (A)) ∪ cl(intθ (A))
and hence by Lemma 2.1,
z-cl(A) ⊂z-cl(int(clθ (A)) ∪ cl(intθ (A)))
=θ-scl[int(clθ (A)) ∪ cl(intθ (A))] ∩ θ-pcl[int(clθ (A)) ∪ cl(intθ (A))]
=[{int(clθ (A)) ∪ cl(intθ (A))} ∪ int(clθ {int(clθ (A)) ∪ cl(intθ (A))})] ∩
[{int(clθ (A)) ∪ cl(intθ (A))} ∪ cl(intθ {int(clθ (A)) ∪ cl(intθ (A))})]
⊂[{int(clθ (A)) ∪ cl(intθ (A))} ∪ int{clθ (int(clθ (A))) ∪ clθ (cl(intθ (A)))}] ∩
[{int(clθ (A)) ∪ cl(intθ (A))} ∪ cl{int(clθ (A)) ∪ cl(intθ (A))}]
⊂[{int(clθ (A)) ∪ cl(intθ (A))} ∪ {int(clθ (A)) ∪ clθ (cl(intθ (A)))}] ∩
[{int(clθ (A)) ∪ cl(intθ (A))} ∪ {cl(int(clθ (A))) ∪ cl(cl(intθ (A)))}]
⊂[{int(clθ (A)) ∪ cl(intθ (A))} ∪ (int(clθ (A)) ∪ cl(intθ (A)))] ∩
[cl(int(clθ (A))) ∪ cl(intθ (A))]
=int(clθ (A)) ∪ cl(intθ (A))
Since A ⊂ z-cl(A), we have z-cl(A) ⊂ int(clθ [z-cl(A)]) ∪ cl(intθ [z-cl(A)]). This
shows that z-cl(A) is a z-open set. On the other hand, z-cl(A) is always a
z-closed set. Therefore z-cl(A) is a z-regular set.
Sufficiency. Let z-cl(A) ∈ zR(X). Then by Theorem 4.10 in [14] we have
A ⊂ z-cl(A) ⊂ int(clθ (z-cl(A))) ∪ cl(intθ (z-cl(A)))
= int(clθ (A)) ∪ cl(int(cl(intθ (A))))
= int(clθ (A)) ∪ cl(intθ (A)).
Hence we have A ∈ zO(X).
358
Murad Özkoç
Theorem 3.3. For a subset A of a topological space X, the following are
equivalent:
(a) A ∈ zR(X),
(b) A = z-cl(z-int(A)),
(c) A = z-int(z-cl(A)).
Proof. (a) ⇒ (b) : Let A ∈ zR(X). Then A = z-int(A) = z-cl(A) and hence
A = z-cl(z-int(A)).
(b) ⇒ (c) : Let A = z-cl(z-int(A)). Then
z-cl(A) = z-cl(z-cl(z-int(A))) ⇒ z-cl(A) = z-cl(z-int(A))
⇒ z-int(z-cl(A)) = z-int(z-cl(z-int(A)))
⇒ z-int(z-cl(A)) = z-int(A) . . . (1)
z-cl(A) ∈ zC(X) ⇒ z-int(z-cl(A)) ∈ zR(X)
⇒ z-cl(z-int(z-cl(A))) = z-int(z-cl(A))
⇒ z-cl(A) = z-int(z-cl(A) . . . (2)
(1), (2) ⇒ z-int(A) = z-cl(A) = z-int(z-cl(A)) ⇒ A = z-int(z-cl(A).
(c) ⇒ (a) : Let A = z-cl(z-int(A)). Since z-int(A) is z-open, by Theorem
3.1(a) we have z-cl(z-int(A)) ∈ zR(X). Hence from hypothesis A ∈ zR(X).
Definition 3.4. A point x of X is called a z-θ-cluster point of A if zcl(U) ∩ A = ∅ for every U ∈ zO(X, x). The set of all z-θ-cluster points of A
is called z-θ-closure of A and is denoted by z-clθ (A). A subset A is said to
be z-θ-closed if A = z-clθ (A). The complement of a z-θ-closed set is said to
be z-θ-open. The family of all z-θ-open (resp. z-θ-closed) sets in X will be
denoted by zθO(X) (resp. zθC(X)).
Theorem 3.5. For a subset A of a space X, the following properties hold:
(a) If A ∈ zO(X), then z-cl(A) = z-clθ (A),
(b) A ∈ zR(X) if and only if A is both z-θ-open and z-θ-closed.
Proof. (a) Generally we have z-cl(A) ⊂ z-clθ (A) for every subset A of X. Let
A ∈ zO(X) and suppose that x ∈
/ z-cl(A). Then there exists U ∈ zO(X, x)
such that U ∩ A = ∅. Since A ∈ zO(X), we have z-cl(U) ∩ A = ∅. This shows
that x ∈
/ z-clθ (A). Hence we obtain z-cl(A) = z-clθ (A).
(b) Let A ∈ zR(X). Then A ∈ zO(X) and by (a), A = z-cl(A) = z-clθ (A).
Therefore, A is z-θ-closed. Since X\A ∈ zR(X), by the argument above, X\A
is z-θ-closed and hence A is z-θ-open. The converse is obvious.
Remark 3.6. It can be easily shown that z-regular ⇒ z-θ-open ⇒ z-open.
But the converses are not necessarily true as shown by the following examples.
On z-θ-open sets and strongly θ-z-continuous functions
359
Example 3.7. Let τ be the usual topology for R and R\N ⊂ R, where N
denotes the set of natural numbers. Then the subset R\N of R is z-θ-open in
R but not z-regular.
Theorem 3.8. Let X be a topological space and A ⊂ X. A is z-θ-open in X
if and only if for each x ∈ A there exists U ∈ zO(X, x) such that z-cl(U) ⊂ A.
Proof. Necessity. Let A ∈ zθO(X) and x ∈ A. Then X\A ∈ zθC(X) and
X\A = z-clθ (X\A). Hence x ∈
/ z-clθ (X\A). Therefore there exists U ∈
zO(X, x) such that z-cl(U) ∩ (X\A) = ∅ and so z-cl(U) ⊂ A.
Sufficiency.
Let A ⊂ X and x ∈ A. From hypothesis there exists U ∈ zO(X, x) such that
z-cl(U) ⊂ A. Therefore z-cl(U) ∩ (X\A) = ∅. Hence X\A = z-clθ (X\A) and
A ∈ zθO(X).
Theorem 3.9. For any subset A of a space X, we have
z-clθ (A) = ∩{V |(A ⊂ V )(V ∈ zθC(X))}
= ∩{V |(A ⊂ V )(V ∈ zR(X))}.
Proof. We prove only the first equality since the other is similarly proved.
First, suppose that x ∈
/ z-clθ (A). Then there exists V ∈ zO(X, x) such that
z-cl(V ) ∩ A = ∅. By Theorem 3.1, X\z-cl(V ) is z-regular and hence X\zcl(V ) is a z-θ-closed set containing A and x ∈
/ X\z-cl(V ). Therefore, we have
x∈
/ ∩{V |(A ⊂ V )(V ∈ zθC(X))}. Conversely, suppose that x ∈
/ ∩{V |(A ⊂
V )(V ∈ zθC(X))}. There exists a z-θ-closed set V such that A ⊂ V and
x ∈
/ V . There exists U ∈ zO(X) such that x ∈ U ⊂ z-cl(U) ⊂ X\V .
Therefore, we have z-cl(U) ∩ A ⊂ z-cl(U) ∩ V = ∅. This shows that x ∈
/ zclθ (A).
Theorem 3.10. Let A and B be any subsets of a space X. Then the following properties hold:
(a) x ∈ z-clθ (A) if and only if U ∩ A = ∅ for each U ∈ zR(X, x),
(b) If A ⊂ B, then z-clθ (A) ⊂ z-clθ (B),
(c) z-clθ (z-clθ (A)) = z-clθ (A),
(d) If Aα is z-θ-closed in X for each α ∈ Λ, then ∩α∈Λ Aα is z-θ-closed in X.
Proof. Clear
Corollary 3.11. Let A and Aα (α ∈ Λ) be any subsets of a space X. Then
the following properties hold:
(a) A is z-θ-open in X if and only if for each x ∈ A there exists U ∈ zR(X, x)
such that x ∈ U ⊂ A,
(b) z-clθ (A) is z-θ-closed,
(c) If Aα is z-θ-open in X for each α ∈ Λ, then ∪α∈Λ Aα is z-θ-open in X.
360
4
Murad Özkoç
Strongly θ-z-continuous Functions and Some
Properties
In this section, we introduce a new type of continuous functions and look into
some relations with some other types.
Definition 4.1. A function f : X → Y is said to be strongly θ-z-continuous
(briefly, st.θ.z.c.) if for each x ∈ X and each open set V of Y containing f (x),
there exists a z-open set U of X containing x such that f [z-cl(U)] ⊂ V .
Definition 4.2. A function f : X → Y is said to be strongly θ-continuous
(briefly, st.θ.c.) [9] (resp. strongly θ-semicontinuous (briefly, st.θ.s.c.) [7],
strongly θ-precontinuous (briefly, st.θ.p.c.) [12], strongly θ-b-continuous (briefly,
st.θ.b.c.) [16], strongly θ-β-continuous (briefly, st.θ.β.c.) [13], strongly θ-e-continuous
(briefly, st.θ.e.c.) [15]) if for each x ∈ X and each open set V of Y containing
f (x), there exists an open (resp. semi-open, preopen, b-open, semi-preopen,
e-open) set U of X containing x such that f [cl(U)] ⊂ V (resp.f [scl(U)] ⊂
V, f [pcl(U)] ⊂ V, f [bcl(U)] ⊂ V, f [spcl(U)] ⊂ V, f [e-cl(U)] ⊂ V ).
Remark 4.3. From Definitions 4.1 and 4.2, we have the following diagram:
st.θ.p.c → st.θ.e.c
↓
st.θ.c
st.θ.z.c
st.θ.b.c → st.θ.β.c
st.θ.s.c
However, none of these implications is reversible as shown by the following
examples.
Example 4.4. Let X = {a, b, c, d, e} and τ = {∅, X, {a} , {c} , {a, c} , {c, d} ,
{a, c, d}} and σ = {∅, X, {d}}. Then the identity function f : (X, τ ) → (X, σ)
is st.θ.z.c. but neither st.θ.b.c. nor st.θ.β.c.
Example 4.5. Let X = {a, b, c, d, e} and τ = {∅, X, {a}, {c}, {a, c}, {c, d},
{a, c, d}} and σ = {∅, X, {b, c, d}}. Then the identity function f : (X, τ ) →
(X, σ) is st.θ.z.c. but not st.θ.e.c.
Example 4.6. Let τ be the usual topology for R and σ = {[0, 1] ∪ ((1, 2) ∩ Q), ∅, R} ,
where Q denotes the set of rational numbers. Then the identity function
f : (R, τ ) → (R, σ) is st.θ.z.c. but neither st.θ.p.c. nor st.θ.s.c.
Example 4.7. Let τ be the usual topology for R and σ = {∅, R, [0, 1) ∩ Q} .
Then the identity function f : (R, τ ) → (R, σ) is st.θ.β.c. but not st.θ.z.c.
On z-θ-open sets and strongly θ-z-continuous functions
361
Theorem 4.8. For a function f : X → Y , the following are equivalent:
(a) f is strongly θ-z-continuous,
(b) For each x ∈ X and each open set V of Y containing f (x), there exists
U ∈ zR(X, x) such that f [U] ⊂ V ,
(c) f −1 [V ] is z-θ-open in X for each open set V of Y ,
(d) f −1 [F ] is z-θ-closed in X for each closed set F of Y ,
(e) f [z-clθ (A)] ⊂ cl(f [A]) for each subset A of X,
(f ) z-clθ (f −1 [B]) ⊂ f −1 [cl(B)] for each subset B of Y .
Proof. (a) ⇒ (b): It follows from Theorem 3.1(a).
(b) ⇒ (c): Let V be any open set of Y and x ∈ f −1 [V ]. There exists U ∈
zR(X, x) such that f [U] ⊂ V . Therefore, we have x ∈ U ⊂ f −1 [V ]. Hence by
Corollary 3.1(a), f −1 [V ] is z-θ-open in X.
(c) ⇒ (d): This is obvious.
(d) ⇒ (e): Let A be any subset of X. Since cl(f [A]) is closed in Y , by (c)
f −1 [cl(f [A])] is z-θ-closed and we have
z-clθ (A) ⊂ z-clθ (f −1 [f [A]]) ⊂ z-clθ (f −1 [cl(f [A])]) = f −1 [cl(f [A])].
Therefore, we obtain f [z-clθ (A)] ⊂ cl(f [A]).
(e) ⇒ (f ): Let B be any subset of Y. By (e), we obtain f [z-clθ (f −1 [B])] ⊂
cl(f [f −1 [B]]) ⊂ cl(B) and hence z-clθ (f −1 [B]) ⊂ f −1 [cl(B)].
(f ) ⇒ (a): Let x ∈ X and V be any open set of Y containing f (x). Since
Y \V is closed in Y , we have z-clθ (f −1 [Y \V ]) ⊂ f −1 [cl(Y \V )] = f −1 [Y \V ].
Therefore, f −1 [Y \V ] is z-θ-closed in X and f −1 [V ] is a z-θ-open set containing
x. There exists U ∈ zO(X, x) such that z-cl(U) ⊂ f −1 [V ] and hence f [zcl(U)] ⊂ V . This shows that f is st.θ.z.c.
Theorem 4.9. Let Y be a regular space. Then f : X → Y is st.θ.z.c. if
and only if f is z-continuous.
Proof. Let x ∈ X and V an open set of Y containing f (x). Since Y is regular,
there exists an open set W such that f (x) ∈ W ⊂ cl(W ) ⊂ V . If f is zcontinuous, there exists U ∈ zO(X, x) such that f [U] ⊂ W . We shall show
that f [z-cl(U)] ⊂ cl(W ). Suppose that y ∈
/ cl(W ). There exists an open set G
containing y such that G ∩ W = ∅. Since f is z-continuous, f −1 [G] ∈ zO(X)
and f −1 [G] ∩ U = ∅ and hence f −1 [G] ∩ z-cl(U) = ∅. Therefore, we obtain
G ∩ f [z-cl(U)] = ∅ and y ∈
/ f [z-cl(U)]. Consequently, we have f [z-cl(U)] ⊂
cl(W ) ⊂ V . The converse is obvious.
Definition 4.10. A space X is said to be z-regular if for each closed set F
and each point x ∈ X\F , there exist disjoint z-open sets U and V such that
x ∈ U and F ⊂ V .
362
Murad Özkoç
Lemma 4.11. For a space X the following are equivalent:
(a) X is z-regular,
(b) For each point x ∈ X and for each open set U of X containing x, there
exists V ∈ zO(X) such that x ∈ V ⊂ z-cl(V ) ⊂ U.
Theorem 4.12. A continuous function f : X → Y is st.θ.z.c. if and only
if X is z-regular.
Proof. Necessity. Let f : X → X be the identity function. Then f is continuous and st.θ.z.c. by our hypothesis. For any open set U of X and any
point x ∈ U, we have f (x) = x ∈ U and there exists V ∈ zO(X, x) such that
f [z-cl(V )] ⊂ U. Therefore, we have x ∈ V ⊂ z-cl(V ) ⊂ U. It follows from
Lemma 4.8 that X is z-regular.
Sufficiency. Suppose that f : X → Y is continuous and X is z-regular. For
any x ∈ X and open set V containing f (x), f −1 [V ] is an open set containing
x. Since X is z-regular, there exists U ∈ zO(X) such that x ∈ U ⊂ zcl(U) ⊂ f −1 [V ]. Therefore, we have f [z-cl(U)] ⊂ V . This shows that f is
st.θ.z.c.
Theorem 4.13. Let f : X → Y be a function and g : X → X × Y be the
graph function of f . If g is st.θ.z.c., then f is st.θ.z.c. and X is z-regular.
Proof. First, we show that f is st.θ.z.c. Let x ∈ X and V an open set of
Y containing f (x). Then X × V is an open set of X × Y containing g(x).
Since g is st.θ.z.c., there exists U ∈ zO(X, x) such that g[z-cl(U)] ⊂ X × V .
Therefore, we obtain f [z-cl(U)] ⊂ V . Next, we show that X is z-regular. Let
U be any open set of X and x ∈ U. Since g(x) ∈ U × Y and U × Y is open in
X × Y , there exists G ∈ zO(X, x) such that g[z-cl(G)] ⊂ U × Y . Therefore,
we obtain x ∈ G ⊂ z-cl(G) ⊂ U and hence X is z-regular.
Theorem 4.14. Let f : X → Y and g : Y → Z be functions. If f is
st.θ.z.c. and g is continuous, then the composition g ◦ f : X → Z is st.θ.z.c.
Proof. It is clear from Theorem 4.6.
We recall that a space X is said to be submaximal [17] if each dense subset
of X is open in X. It is shown in [17] that a space X is submaximal if and
only if every preopen set of X is open. A space X is said to be extremally
disconnected [5] if the closure of each open set of X is open. A space X is
said to be regular [5] if for each closed set F and each point x ∈ X\F , there
exist disjoint open sets U and V such that x ∈ U and F ⊂ V . Note that an
extremally disconnected space is exactly the space where every semiopen set
is α-open.
On z-θ-open sets and strongly θ-z-continuous functions
363
Theorem 4.15. Let X be a regular submaximal extremally disconnected
space. Then the following properties are equivalent for a function f : X → Y :
(a) f is st.θ.c. (b) f is st.θ.s.c. (c) f is st.θ.p.c. (d) f is st.θ.b.c. (e) f is
st.θ.e.c. (f ) f is st.θ.z.c. (g) f is st.θ.β.c.
Proof. It follows from the fact that if X is regular submaximal extremally
disconnected, then open set, preopen set, semiopen set, b-open set, e-open set,
z-open set and β-open set are equivalent.
5
Separation Axioms
Definition 5.1. A space X is said to be z-T2 if for each pair of distinct
points x and y in X, there exist U ∈ zO(X, x) and V ∈ zO(X, y) such that
U ∩ V = ∅.
Lemma 5.2. A space X is z-T2 if and only if for each pair of distinct
points x and y in X, there exist U ∈ zO(X, x) and V ∈ zO(X, y) such that
z-cl(U) ∩ z-cl(V ) = ∅.
Theorem 5.3. If f : X → Y is a st.θ.z.c. injection and Y is T0 , then X
is z-T2 .
Proof. For any distinct points x and y of X, by hypothesis f (x) = f (y) and
there exists either an open set V containing f (x) not containing f (y) or an open
set W containing f (y) not containing f (x). If the first case holds, then there
exists U ∈ zO(X, x) such that f [z-cl(U)] ⊂ V . Thus, we obtain f (y) ∈
/ f [zcl(U)] and hence X\z-cl(U) ∈ zO(X, y). If the second case holds, then we
obtain a similar result. Thus, X is z-T2 .
Theorem 5.4. If f : X → Y is a st.θ.z.c. function and Y is Hausdorff,
then the subset A = {(x, y)|f (x) = f (y)} is z-θ-closed in X × X.
Proof. It is clear that f (x) = f (y) for each (x, y) ∈
/ A. Since Y is Hausdorff,
there exist open sets V and W of Y containing f (x) and f (y), respectively,
such that V ∩ W = ∅. Since f is st.θ.z.c., there exist U ∈ zO(X, x) and
G ∈ zO(X, y) such that f [z-cl(U)] ⊂ V and f [z-cl(G)] ⊂ W . Put D = f [zcl(U)] × f [z-cl(G)]. It follows that (x, y) ∈ D ∈ zR(X × X) and D ∩ A = ∅.
This means that z-clθ (A) ⊂ A and thus, A is z-θ-closed in X × X.
We recall that for a function f : X → Y , the subset {(x, f (x))|x ∈ X} of
X × Y is called the graph of f and is denoted by G(f ).
Definition 5.5. The graph G(f ) of a function f : X → Y is said to be
strongly z-closed if for each (x, y) ∈ (X × Y )\G(f ), there exist U ∈ zO(X, x)
and an open set V in Y containing y such that (z-cl(U) × V ) ∩ G(f ) = ∅.
364
Murad Özkoç
Lemma 5.6. The graph G(f ) of a function f : X → Y is strongly z-closed
if and only if for each (x, y) ∈ (X × Y )\G(f ), there exist U ∈ zO(X, x) and
an open set V in Y containing y such that f [z-cl(U)] ∩ V = ∅.
Theorem 5.7. If f : X → Y is st.θ.z.c. and Y is Hausdorff, then G(f ) is
strongly z-closed in X × Y .
Proof. It is clear that f (x) = y for each (x, y) ∈ (X × Y )\G(f ). Since Y
is Hausdorff, there exist open sets V and W in Y containing f (x) and y,
respectively, such that V ∩ W = ∅. Since f is st.θ.z.c., there exists U ∈
zO(X, x) such that f [z-cl(U)] ⊂ V . Thus, f [z-cl(U)] ∩ W = ∅ and then by
Lemma 5.4, G(f ) is strongly z-closed in X × Y .
6
Covering Properties
Definition 6.1. A space X is said to be
(a) z-closed if for every z-open cover {Vα |α ∈ A} of X, there exists a finite
subset A0 of A such that X = {z-cl(Vα )|α ∈ A0 };
(b) countably z-closed if for every countable z-open cover {Vα |α ∈ A} of X,
there exists a finite subset A0 of A such that X = {z-cl(Vα )|α ∈ A0 }.
A subset A of a space X is said to be z-closed relative to X if for every
cover {Vα : α ∈ Λ} of A by z-open sets of X, there exists a finite subset Λ0 of
Λ such that A ⊂ ∪{z-cl(Vα ) : α ∈ Λ0 }.
Theorem 6.2. If f : X → Y is a st.θ.z.c. function and A is z-closed
relative to X, then f [A] is a compact set of Y .
Proof. Let {Vα : α ∈ Λ} be a cover of f [A] by open sets of Y . For each point
x ∈ A, there exists α(x) ∈ Λ such that f (x) ∈ Vα(x) . Since f is st.θ.z.c., there
exists Ux ∈ zO(X, x) such that f [z-cl(Ux )] ⊂ Vα(x) . The family {Ux : x ∈ A}
is a cover of A by z-open sets of X and hence there exists a finite subset A0
of A such that A ⊂ ∪x∈A0 z-cl(Ux ). Therefore, we obtain f [A] ⊂ ∪x∈A0 Vα(x) .
This shows that f [A] is compact.
Corollary 6.3. Let f : X → Y be a st.θ.z.c. surjection. Then the following
properties hold:
(a) If X is z-closed, then Y is compact.
(b) If X is countably z-closed, then Y is countably compact.
Theorem 6.4. If a function f : X → Y has a strongly z-closed graph, then
f [A] is closed in Y for each subset A which is z-closed relative to X.
On z-θ-open sets and strongly θ-z-continuous functions
365
Proof. Let A be z-closed relative to X and y ∈ Y \f [A]. Then for each x ∈ A
we have (x, y) ∈
/ G(f ) and by Lemma 5.4 there exist Ux ∈ zO(X, x) and an
open set Vx of Y containing y such that f [z-cl(Ux )] ∩ Vx = ∅. The family
{Ux : x ∈ A} is a cover of A by z-open sets of X. Since A is z-closed relative
to X, there exists a finite subset A0 of A such that A ⊂ ∪{z-cl(Ux ) : x ∈
A0 }. Put V = ∩{Vx : x ∈ A0 }. Then V is an open set containing y and
f [A] ∩ V ⊂ [∪x∈A0 f [z-cl(Ux )]] ∩ V ⊂ ∪x∈A0 [f [z-cl(Ux )] ∩ Vx ] = ∅. Therefore,
we have y ∈
/ cl(f [A]) and hence f [A] is closed in Y .
Theorem 6.5. Let X be a submaximal extremally disconnected space. If a
function f : X → Y has a strongly z-closed graph, then f −1 [A] is θ-closed in
X for each compact set A of Y .
Proof. Let A be a compact set of Y and x ∈
/ f −1 [A]. Then for each y ∈ A
we have (x, y) ∈
/ G(f ) and by Lemma 5.4 there exists Uy ∈ zO(X, x) and an
open set Vy of Y containing y such that f [z-cl(Uy )] ∩ Vy = ∅. The family
{Vy : y ∈ A} is an open cover of A and there exists a finite subset A0 of A
such that A ⊂ ∪y∈A0 Vy . Since X is submaximal extremally disconnected, each
Uy is open in X and z-cl(Uy ) = cl(Uy ). Set U = ∩y∈A0 Uy , then U is an open
set containing x and
f [cl(U)] ∩ A ⊂ ∪x∈A0 [f [cl(U)] ∩ Vy ] ⊂ ∪x∈A0 [f [z-cl(Uy )] ∩ Vy ] = ∅.
Therefore, we have cl(U) ∩ f −1 [A] = ∅ and hence x ∈
/ clθ (f −1 [A]). This shows
that f −1 [A] is θ-closed in X.
Corollary 6.6. Let X be a submaximal extremally disconnected space and
Y be a compact Hausdorff space. For a function f : X → Y, the following
properties are equivalent:
(a) f is st.θ.z.c.,
(b) G(f ) is strongly z-closed in X × Y ,
(c) f is strongly θ-continuous,
(d) f is continuous,
(e) f is z-continuous.
Proof. (a) ⇒ (b). It follows from Theorem 5.5.
(b) ⇒ (c). It follows from Theorem 6.4.
(c) ⇒ (d) ⇒ (e). These are obvious.
(e) ⇒ (a). Since Y is regular, it follows from Theorem 4.7.
This study is dedicated to Professor Zekeriya GÜNEY on the occasion of
his 65th birthday.
366
Murad Özkoç
References
[1] M.E. Abd El-Monsef, S.N. El-Deeb and R.A. Mahmoud, β-open sets and
β-continuous mapping, Bull. Fac. Sci. Assiut Univ. 12 (1983), 77-90.
[2] T. Altunöz and G. Aslım, On new decompositions of continuity, J. Adv.
Res. Pure Math. (JARPM) 3 (2011), no.2, 84-92.
[3] D. Andrijević, Semi-preopen sets, Mat. Vesnik 38 (1986), 24-42.
[4] D. Andrijević, On b-open sets, Mat. Vesnik 48 (1996), 59-64.
[5] N. Bourbaki, General Topology, Part I Addison Wesley, Mass., 1996.
[6] E. Ekici, On e-open sets, DP ∗ -sets and DPE ∗ -sets and decompositions of
continuity, Arab. J. Sci. Eng. 33 (2008), number 2A,269-282.
[7] S. Jafari and T. Noiri, On strongly θ-semi-continuous functions, Indian
J. Pure Appl. Math. 29 (1998), 1195 - 1121.
[8] N. Levine, Semi-open sets and semi-continuity in topological spaces,
Amer. Math. Monthly 70 (1963), 36-41.
[9] P.E. Long and L.L. Herrington, Strongly θ-continuous functions, J. Korean Math. Soc. 18 (1981/82), no.1, 21-28.
[10] A.S. Mashhour, M.E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and weak precontinuous functions, Proc. Math. Phys. Soc. Egypt 53
(1982), 47 - 53.
[11] O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15
(1965), 961 - 970.
[12] T. Noiri, On strongly θ-precontinuous functions, Acta Math. Hungar. 90
(2001), 307 - 316.
[13] T. Noiri and V. Popa, Strongly θ-β-continuous functions, J. Pure Math.
19 (2002), 31 - 39.
[14] M. Özkoç, On z-open sets and z-continuous functions (submitted).
[15] M. Özkoç and G. Aslım, On strongly θ-e-continuous functions, Bull. Korean Math. Soc. 47(5) (2010), 1025-1036.
[16] J.H. Park, Strongly θ-b-continuous functions, Acta Math. Hungar. 110(4)
(2006), 347-359.
On z-θ-open sets and strongly θ-z-continuous functions
367
[17] I.L. Reilly and M.K. Vamanamurthy, On some questions concerning preopen sets, Kyungpook Math. J. 30 (1990), 87-93.
[18] N.V. Velicko, H-closed topological spaces, Amer. Math. Soc. Transl. 78
(1968), 103-18.
Received: January 9, 2013