Download Syndetic Sets

Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(24): 2016
ON -Syndetic Sets
Aqeel ketab Mezaal Al-khafaji
Department of Computer Sciences, College of Sciences for women Babylon
University,
[email protected]
Abstract
In this paper, we introduce and investigate the notion of -syndetic sets, using the notion of -open
sets which was introduce by Njastad. And the work of W. Gottschalk , and S. Al-Kutaibi .
Keywords: -open, −
, −closure, −
, −
‫ﺍﻟﺨﻼﺼﺔ‬
‫ﻓﻲ ﻫﺫﺍ ﺍﻟﺒﺤﺙ ﻗﺩﻤﻨﺎ ﻤﻔﻬﻭﻡ ﺍﻟﻤﺠﻤﻭﻋﺎﺕ ﺍﻟﺭﺍﺒﻁﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ ﻤﻊ ﻋﺩﺩ ﻤﻥ ﺍﻟﻨﻅﺭﻴﺎﺕ ﺒﺎﺴﺘﺨﺩﺍﻡ ﺘﻌﺭﻴﻑ ﺍﻟﻤﺠﻤﻭﻋﺎﺕ ﺍﻟﻤﻔﺘﻭﺤﺔ‬
.‫ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ ﺍﻟﺘﻲ ﻗﺩﻤﻬﺎ ﻨﺠﺎﺴﺘﺩ ﻭﻤﻌﺘﻤﺩﻴﻥ ﺒﺫﻟﻙ ﻋﻠﻰ ﺍﻟﻤﻔﺎﻫﻴﻡ ﺍﻟﺘﻲ ﻗﺩﻤﻬﺎ ﻜل ﻤﻥ ﺠﻭﺘﺴﺠﻭﻙ ﻭﺍﻟﻜﺘﺒﻲ ﻓﻲ ﻫﺫﺍ ﺍﻟﻤﻭﻀﻭﻉ‬
‫ ﺍﻟﻤﺠﻤﻭﻋﺔ‬،‫ ﺍﻻﻨﻐﻼﻕ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ‬،‫ﺍﻟﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻐﻠﻘﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ‬،‫ ﺍﻟﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﻔﺘﻭﺤﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ‬:‫ﺍﻟﻜﻠﻤﺎﺕ ﺍﻻﻓﺘﺘﺎﺤﻴﺔ‬
.‫ ﺍﻟﻤﺠﻤﻭﻋﺔ ﺍﻟﻤﺘﺭﺍﺒﻁﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ‬،‫ﺍﻟﻤﺭﺼﻭﺼﺔ ﻤﻥ ﺍﻟﻨﻭﻉ ﺍﻟﻔﺎ‬
1.Introduction
Gottschalk in [Gottschalk, 1955] introduced the notions of left (right) syndetic set
in topological group. He defined a subset
of topological group , is called left
)=
(right) syndetic if there exists a compact subset
of , such that
= (
. In 1998 Al-kutaibi [Al-kutaibi, 1998] introduced paper semi-syndetic and feebly
syndetic subsets.
The aim of this paper is to introduce and investigate a new class of syndetic called
−
which is a subset
of topological group , is called left (right)
)=
−
if there exists a −
set of , such that
= (
. And we obtain several Theorem abut −
.
2. Preliminaries
In this section we recall some of the basic Definitions.
Throughout this paper will denote a topological space. If A is a subset of the space
( ), − ( ) denote the closure, interior and the −closure of A
, ( ),
respectively. A subset of is said to be −
set [Njastad, 1965] if A ⊂
Int(C1(Int(A))). The complement of a −
set is said to be −
. Let
⊆ and ℋ be a family of
−
subsets of , then ℋ is called a −open
cover of , if ⊆ ⋃ ∈ℋ , A is said to be −
if and only if any −
cover has a finite sub cover.
Clear that, every open set is −
and then every −
set is compact
also, the union of two −
subsets of
is −
. A topological
group is a set which carries group stricture and a topology and satisfied the two
axioms: (i) The map ( , ) ⟶
of × into
is continuous.(That is, the
operation of is continuous). (ii) The map ⟶
(The inversion map) of into
is continuous.[Higgins, 1979]
3. The main results
3.1 Theorem:
If is −
set in a topological group
329
, then
is
−
.
Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(24): 2016
Proof:
Let : ⟶ be the invers map, that is, ( ) =
for all in , let ℋ be a
)=(
−
cover of
, then (ℋ ) is −
cover of (
) =
,but
is −
, which implies (ℋ ) has a finite sub cover ℋ ∗ , then
(ℋ ∗ ) covers ( ) =
. Hence,
is −
set.
3.2 Definition:
Let
be a subset of a topological group , then
is called left (right) −
if there exists a
−
subset
of
such that
=
(
= ).
3.3 Note:
In the following results we will prove the case of left −
and the case
of right −
will be similar.
3.4 Theorem:
Let be a topological group and let ⊆ , then is left (right) −
set
in if and only if there exists a
−
subset
of such that every left
(right) translation of intersects .
Proof:
Sufficiency, suppose is a left −
set then, there exists a −
subset of such that
= , let ∈ then, there exists ∈ , ∈ such that
=
which implies =
and then ∈
but
is −
.
Hence
∩ ≠ ∅, (i.e
is the −
set we need).
Efficiency, let ∈ , there exists −
subset
of such that,
∩
≠ ∅, for each in , there exists ∈ , ∈ such that
= , so =
,
which implies =
, and since
is −
then is a left −
.
3.5 Theorem:
Let be a subset of a topological group , then is left (right) −
in
, if and only if
right (left) −
.
Proof:
Let is a left −
, then there exists −
subset
of such
that
= . Since
=
=(
) =
and since
is
−
, then
is right −
.
3.6 Theorem:
Let be a topological group, let , be two subsets of such that
⊆ . If
A is left (right) −
set, then so is .
Proof:
Let be a left −
set, then there exists a −
subset
of
such that
= , since ⊆ , then
= , which implies that
is left −
.
3.7 Theorem:
Let be a topological group, then
1- If is a left −
set in ,then ( )
− ( ) are left
2- The union of any family of left (right) −
sets is left (right)
−
.
Proof:
Directly from (Theorem 3.6).
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Journal of Babylon University/Pure and Applied Sciences/ No.(2)/ Vol.(24): 2016
3.8 Theorem:
Let be a topological group, let , are two left (right) −
sub sets
of , then ∩ is a left (right) −
.
Proof:
Since both
are left −
, then there exist −
sets
such that
=
=
and then ( ∩ )( ∪ ) =
( ∪ ) ∩ ( ∪ ) = ∩ = and since ( ∪ ) is −
, then
( ∩ ) is left −
.
3.9 Theorem:
Let be a subset of topological group .If is a subgroup of or if is an
abelian group, then is left −
in if and only if is a right −
in .
Proof:
Let is a left −
subgroup of , then =
, that means
=
and hence =
, but
is −
, which implies is a
right −
in .
3.10 Theorem:
Let
be a −
subgroup of a topological group
quotient space ∕ is compact.
, then the
Proof:
Let be a
exists a −
map. Clear that (
⊂
, that is
is continuous and
is compact set.
−
subgroup of a topological group , then there
set such that
= . Let : ⟶ ∕ is the quotient
) ⊂ ⁄ . Let
∈ ∕ , then
⊂
which implies
∈ ( ) and then, ⁄ ⊂ ( ). Hence ⁄ = ( ). Since
is −
and hence is compact, then ( ) = ⁄
Refrences
ALkutaibi,S.H., On semi-syndetic and feebly syndetic subsets, Journal of sciences.
College of Education .Tikrit University, Vol 4, No.3, (1998).pp.
Gottschalk, W.H., Hedlund , G.A., Topological dynamics, American Mathematical
Society, (1955)
Higgins, P.J., An introduction to topological groups, Cambridge University Press,
(1979).
Munkres , J.R., Topology, 2nd Ed. , PHI , (2000).
Njåstad O., On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961−970.
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