Abstract-- In this paper, we define and introduce a new type of topological transitive map called θ- transitive and investigate
some of its properties in (X, τθ
),
denotes the θ–topology of agiven topological space (X, τ ). Further, we introduce the notions of θ- minimal mapping. We have proved that every topologically transitive map is a θ-type transitive map but the converse not necessarily true, unless the space X is regular and that every minimal map is a θ- minimal map, but the converse not necessarily true.
Index Term-- Topologically θ- transitive, θ- minimal maps, θ- irresolute, θ- dense.
1. INT RODUCT ION
In 1943, Fomin [10] introduced the notion of θ-continuous maps. The notions of θ-open sets, θ-closed sets and θ-closure where introduced by Veliˇcko [1] for the purpose of studying the important class of H-closed spaces in terms of arbitrary fiber-bases. Dickman and Porter [2], [3], Joseph [4] and Long and Herrington [14] continued the work of Velicˇko. Recently, Jafari [6] has also obtained several new and interesting results related to these sets. For these sets and maps, we introduce the notions of θ-type transitive maps, θ-minimal maps and show that some of their properties are analogous to those for topologically transitive maps. Also, we give some additional properties of θ-irresolute maps. We denote the interior and the closure of a subset A of X by Int(A) and Cl(A), respectively. By a space X, we mean a topological space (X, τ ) A point x ∈
X is called a θ-adherent point of A [1], if
A
Cl
(
V
)
for every open set V containing x. The set of all θ-adherent points of a subset A of X is called the θ-closure of A and is denotedby
Cl
(
A
)
.)
(
if
closed
-called
is
X
of
A
subset
A
A
Cl
A
. Dontchev and Maki [4] have shown that if A and B are
subsets of a space (X, τ ), then
(B) Cl (A) Cl = B) (A
Cl
and Cl(A B)=Cl (A)Cl(B). Note also that the
θ-closure of a given set need not be a
-
closed
set
. But it is.
always closed. Dickman and Porter [2] proved that a compact subspace of a Hausdorff space is θ-closed. Moreover, they showed that a θ-closed subspace of a Hausdorff space is closed. Jankovi´[7] proved that a space (X, τ ) is Hausdorff if and only if every compact set is θ-closed. The complement of a θ-closed set is called a θ-open set. The family of all θ-open sets forms a topology on X and is denoted by
pology.
to
or
This topology is coarser than τ and
that a space (X, τ ) is regular if and only if
[8]. Then we observe that every theta-type transitive maps is transitive if)
,
(
X
is regular. Recall that a point x ∈ X is called the δ-cluster point of A ⊆ X if A ∩ Int(Cl(U )) = ∅ for every open set U of X containing x. The set of all δ-cluster points of A is called the δ-closure of A, denoted by Clδ (A). A subset A ⊆ Xis called δ-closed if A = Clδ (A). The complement of a δ -closed
set is called δ-open. Similarly, the δ -interior of a set A in X, written
Int
(
A
)
, consists of those points x of A such that for some regularly open set U containing x,U
A
. A set Ais
-
open
iff
Int
(
A
)
A
, or equivalently, X\A is δ-closed. It is worth to be noticed that the family of all δ -open subsets of (X, τ ) is a topology on X which is denoted by τδ . Itshould be noted that
Cl
(
A
)
is the closure of A with respect to(
X
,
)
. In general,Cl
(
A
)
will not be the closure of A with respect to(
X
,
)
. It is easily seen that one always has
A Cl A A
Cl A Cl
A ( ) ( ) ( ) where A denotes
the closure of A with respect to
(
X
,
)
. It is also obviousthat a set A is θ-closed in (X, τ ) if and only if it is closed in
)
,
(
X
. The space(
X
,
)
is called sometimes the semi regularization of (X, τ ). A functionf
:
X
Y
is closure continuous (θ-continuous) at x ∈ X if given any open set V in Y containing f(x), there exists an open set U in X containing x such thatf
(
Cl
(
U
))
Cl
(
V
).
[12]In this paper, we will define a new class of topological transitive maps called
type
transitive
and a newclass of
minimal
maps
. We have shown that everyIntroduction to
-Type Transitive Maps on
Topological spaces
Mohammed Nokhas Murad Kaki
Math Department, School of Science Faculty of Science and Science Education,transitive map is a θ-type transitive map, but the converse not necessarily true and that every minimal map is a θ-minimal map, but the converse not necessarily true we will also study some of their properties.
2. BASIC DEFINIT IONS AND THEOREMS
Theorem 2.1 [13] For subsets A, B of a space X, the following stateme-
nts hold: (1)
A
of
set
derived
the
is
A
D
where
A
D
A
D
(
)
(
)
(
)
(2)
If
A
B
,
then
D
(
A
)
D
(
B
)
(3)
)
(
)
(
)
(
)
(
)
(
)
(
A
D
B
D
A
B
and
D
A
B
D
A
D
B
D
Note that the family
of θ –open sets in(
X
,
)
always forms a topology on X denoted θ -topology and that θ-topology coarser than τ.Definition 2.2 [15]: Let A be a subset of a space X. A point x is said to be an
- limit point of A if for each
- open U containing x,U
(
A
\
x
)
.The set of all
- limit points of A is called the
-derived set of A and is denoted by)
(
A
D
.Definition 2.3 [15] For subsets A and B of a space X, the following statements hold true:
1)
D
(
A
)
D
(
A
)
where D(A) is the derived set of A 2) ifA
B
thenD
(
A
)
D
(
B
)
3)
D
(
A
)
D
(
B
)
D
(
A
B
)
4)
D
(
A
D
(
A
))
A
D
(
A
)
.It is clear that
D
(
A
)
D
(
A
)
D
(
A
)
D
(
A
)
Theorem 2.4.[13] For subsets A, B of a topological space X, the following statements are
true:
)
(
)
1
(
Int
A
is the union of all open sets of X whose closure are contained in AA
A
Int
open
is
A
(
)
)
2
(
.)
(
))
(
(
)
3
(
Int
Int
A
Int
A
.)
\
(
)
(
\
)
4
(
X
Int
A
Cl
X
A
.)
\
(
)
(
\
)
5
(
X
Cl
A
Int
X
A
.
(
6
)
If
A
B
then
Int
(
A
)
Int
(
B
)
.(
7
)
Int
(
A
)
Int
(
B
)
Int
(
A
B
)
(
8
)
Int
(
A
)
Int
(
B
)
Int
(
A
B
)
.3. TRANSIT IVIT Y AND MINIMAL SYST EMS
Topological transitivity is a global characteristic of dynamical systems.
By a system (X,f) short for a dynamical system (X,f)
[17] we mean a
topological space X together with a continuous map
X
X
f
:
. Thespace X is sometimes called the phase space of the system )
,
(X f . A s
subset
A
of
X
is called f inveriant ifA
A
f
(
)
.Definition 3.1. A system (X,f) is called minimal if X does not contain
any non-empty,
proper, closed f inveriant subset. In such a case we also say that the map
f itself is minimal.
Given a point x in a system (X, f ) ,
...}
(x),
f
(x),
f
{x,
(x)
O
f
2orbit we mean a forward orbit even if f is a homeomorphism) and
f(
x
)
ω -limit set, i.e. the set of limit pointsof the sequence
x,
f
(x),
f
2(x),
...
The following conditions are equivalent:
X, f ) is minimal,
X
x
f
(
)
for every x ∈ X .A minimal map f is necessarily surjective if X is assumed to be Hausdorff and compact. Now, we will study the Existence of minimal sets. Given a dynamical system (X,f) , a set
A
X
is called a minimal set if it is non-empty, closed and invariant and if no proper subset of A has these three properties. So,
X
A
is a minimal set if and only if (A, f |A) is a minimal system. A system (X,f) is minimal if and only if X is aminimal set in (X,f) . The basic fact discovered by G. D.
Birkhoff is that in any compact system (X,f) there are
minimal sets. This follows immediately from the Zorn's lemma. Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. This is how compact minimal sets may appear in non-compact spaces. Two minimal sets in (X,f) either are disjoint or coincide. A minimal set A
is strongly f inveriant , i.e. f (A A. Provided it is compact Hausdorff
Definition 3.2. A system (X,f) is called topologically mixing if for any pair
U, V of non-empty open sets there exists
N
such that forall n ≥ N we have
f
n(
U
)
V
.Topologically mixing conveys the idea that each open set U, after iterations of the map f, for each open set V , for all n
sufficiently large,
f
n(
U
)
intersects VDefinition 3.3 Let (X,
) be a topological space, andX
X
topologically transitive map if for every pair of open sets U and V in X there is a positive integer n such that
V
U
f
n(
)
.Definition3.4 Two topological systems
Y
Y
g
and
X
X
f
:
:
are said to be topologicallyconjugate if there is a homeomorphism
h
:
X
Y
such thath
f
g
h
. We will call h a topological conjugacy.Proposition 3.5 if
f
:
X
X
and
g
:
Y
Y
are topologically conjugate . Then(1)
f
is topologically transitive if and only ifg
istopologically transitive;
(2)
f
is minimal if and only ifg
is minimal;(3)
f
is topologically mixing if and only ifg
is topologicallymixing.
4. θ-TYPE TRANSIT IVE MAPS
In this section we generalize topologically transitive maps to θ-type transitive maps that may not be transitive . We define the θ-type transitive maps on a space
(
X
,
)
and θ – type minimal maps that may not be minimal, ), and we study some of their properties and prove some results associated with these new definitions We investigate some properties and characterizations of such functions.Definition 4.1 Let (X, τ) be a topological space. A subset A of X is called θ-dense in X if
Cl
(
A
)
X
.
Remark 4.2 Any θ-dense subset in X intersects any θ -open set in X.
Proof: Let A be a θ-dense subset in X, then by definition,
X
A
Cl
(
)
, and let U be a non-empty θ-open set in X. Suppose that A∩U=ϕ. Therefore cU
B
is θ-closed because B is the complement of θ-open and cU
A
=B. So)
(
A
Cl
Cl
(
B
)
, i.e.Cl
(
A
)
B, butCl
(
A
)
X
, so X
B, this contradicts that U ≠
Definition 4.3. A function
f
:
X
X
is called θ − irresolute if the inverse image of each θ − open set is a θ − open set in X. Definition 4.4 A subset A of a topological space (X,τ) is said to be nowhere θ-dense, if its θ-closure has an empty θ-interior, that is,int
(
Cl
(
A
))
.Definition 4.5 Let (X, τ) be a topological space ,
X
X
f
:
be θ-irresolute map thenf
is said to betopological θ-type transitive if every pair of non-empty θ-open sets U and V in X there is a positive integer n such that
.
)
(
U
V
f
n . Associated with this new definition 4.5 wecan prove the following new theorem.
Theorem 4.6: Let (X, τ) be a topological space and
X
X
f
:
be θ -irresolute map. Then the following statements are equivalent:(1)
f
is θ-type transitive map(2)
(
)
0
f
D
n n
is θ-dense in X, with D is θ-open set in X.(3) ( )
0f D
n n
is θ-dense in X with D is θ-open set in X
(4) If
B
X
is θ-closed andf
(
B
)
B
.
then B=X or B is nowhere θ-dense(5) If
f
1(
D
)
D
and D is open in X then D=ϕ or D is θ-dense in X, we have to prove th theorem.Proof: (1)
(2)Assume that ( )
0f D
n n
is not θ-dense.
Then by Remark 4.2, there exists a non-empty θ-open set E such that
f D E
n n0 ( )
.
This implies that fn(D)E for all n є N .This is a
contradiction to the θ- transitivity of the map f. Hence
) (
0f D
n n
is θ-dense in X.
(2)
(1)Let D and E be two nonempty θ -open sets in X, and let
) (
0f D
n n
be θ-dense in X, this implies that
f D E
n n 0 ( )
by Remark 4.2.Therefore there exists m є N such that
E D fn
)
( . Hence f is a θ-type transitive map.
(1)
(3)It is obvious that
(
)
0
f
D
n n
is θ-open, and since f is θ-type transitive, it has to meet every θ-open set in X, and hence)
(
0
f
D
n n
is θ-dense by Remark 4.2..(3)
(1)Let V and D be two θ-open subsets in X. Then
(
)
0
f
D
n n
isθ-dense, this implies that
f D V
n
n0 ( )
, by Remark 4.2
.This implies that there exists m є N such that
V D
f m( ) ..So fm(fm(D)V) Dfm(V).
Therefore
f
is θ-type transitive.(1)
(4) Letf
be θ-type transitive map,B
X
beθ-closed and
f
(
B
)
B
Assume that B ≠
and B has a nonempty θ-interior (i.e.int
(
B
)
). If we define V=X\B so V is θ-open because V is the compliment of θ-closed. LetB
W
be θ-open sinceint
(
B
)
.
We haveB
W
f
n(
)
since B is invariant. Therefore fnW V )( ,
for all n є N. This is a contradiction to θ-type transitive. Hence B=X or B is nowhere θ-dense.
Let V be a nonempty θ-open set in X. Suppose
f
is not aθ-type transitive map, from (3) of this theorem
(
)
0
f
V
n n
is notθ-dense, but θ-open. Define
\
(
)
0
f
V
X
B
nn
which isθ-closed, because it is the complement of θ-open, and B≠X.
Clearly
f
(
B
)
B
Since(
)
0
f
V
n n
is not θ-dense so byRemark 4.2, there exists a non-empty θ-open W in X such that
.
)
(
0
f
V
W
n
n This implies that
W
B
. This iscontradiction to the fact that B is nowhere θ-dense. Hence the map
f
is θ-type transitive..(1)
(5) Suppose thatf
is θ-type transitive map,D
X
is θ-open and f1(D)D. Assume thatD
and D is notθ-dense in X (i.e.
Cl
(
D
)
X
).
Then there exists a non-empty θ-open V in X such thatD
V
.
Further fn D V) (
for all n є N. This implies
D
f
n(
V
)
for all n є N, a contradiction to f being a θ-type transitive map. Therefore
D
or D is θ-dense in X.5. θ- MINIMAL MAPS
In this section, we introduce a new definition of θ -minimal maps and we study some of its properties.
Given a topological space X, we ask whether there exists θ-continuous map on X such that the set
...}
),
(
...,
),
(
),
(
,
{
x
f
x
f
2x
f
nx
, called the orbit of x and denoted byO
f(
x
)
, is θ-dense in X for each x є X.. A partial answer will be given in this section. Let us begin with a new definitionDefinition 5.1 (θ-minimal) Let X be a topological space and f an θ-continuous map on X. Then (X,f) is called θ -minimal system (or f is called θ-minimal map on X) if one of the three equivalent conditions hold:
1)The orbit of each point of X is θ-dense in X. 2)
Cl
(
O
f(
x
))
X
for each x є X .3)Given x є X and a nonempty θ-open U in X, there exists n є
N such that
f
n(
x
)
U
.Theorem 5.2 For
(
X
,
f
)
the following statements are equivalent:(1)
f
is an θ-minimal map. (2) If E is an θ-closed subset of Xwith
f
(
E
)
E
,
we say E is invariant. Then E=
or E=X. (3) If U is a nonempty θ-open subset of X, thenX
U
f
nn
0
(
)
.Proof: (1)
(2): If A ≠
, let x є A. Since A is invariant and θ-closed,Cl
(
O
f(
x
))
A
. On otherhand
Cl
(
O
f(
x
))
X
.
Therefore,X
A
So A=X(2)
(3) Let A=X\(
)
0
f
U
n n
. Since U is nonempty, A ≠ X. Since U is θ-open andf
is θ-continuous, A is θ-closed. AlsoA
A
f
(
)
, so A must be an empty set.(3)
(1): Let x є X and U be a nonempty θ-open subset of X.Since x є X =
(
)
0
f
U
n n
.Therefore x єf
n(
U
)
for somen>0. So
f
n(
x
)
U
6. CONCLUSION
There are the main results of the paper.
Proposition 6.1 Every transitive map is a θ-type transitive map as every θ-open set is an open set, but the converse not necessarily true.
Proposition 6.2 Every minimal map is a θ-minimal map as every θ-open set is open set, but the converse not necessarily true.
Theorem 6.3 For
(
X
,
f
)
the following statements are equivalent:(1)
f
is an θ-minimal map.(2) If E is an θ-closed subset of X with
f
(
E
)
E
,
we say E is invariant. Then E=
or E=X.(3) If U is a nonempty θ-open subset of X, then
X
U
f
nn
0
(
)
.Theorem 6.4 Let (X, τ) be a topological space and
X
X
f
:
be θ -irresolute map. Then the following statements are equivalent:(1)
f
is θ-type transitive map(2)
(
)
0
f
D
n n
is θ-dense in X, with D is θ-open set in X.(3) ( )
0f D
n n
is θ-dense in X with D is θ-open set in X
(4) If
B
X
is θ-closed andf
(
B
)
B
.
then B=X or B is nowhere θ-dense(5) Iff
1(
D
)
D
and D is θ--open in X then D=ϕ or D is θ-dense in XREFERENCES
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