Ijeas0309022

International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-9, September 2016

  P Conectedness

in L  Bitopological spaces

FangLi Li , Hongkui Li, Tao Li  Abstract— This paper introduce open sets and

L  bts as L  bitopological space and A  as the closure

p  closed

1

of A in L ,  1  6 .

sets into L  Bitopological spaces, and based on this we introduce some related definitions and theorems about   P closed set and

X

B. Final Stage The Related Defines And Conclusion About p  open set And p  close Set

  P open sets . Furthermore, we give a new concept about   P local-connectivity .Then, it point out   P local-connectivity has two properties of topological invariance and finitely productive property and proves the other relevant theories.





Definition 2.2: Let LX ,  be L  bts , A  LX and record as p  open set ,if and only if have open set U ,makes

A  U  A  ;If A is a p  open set, we call A is a

Index Terms— L  bitopological spaces,   p local-connectivity, topological invariance, finitely

p  close set .The whole p  open sets in LX ,   are

  ,and the whole

productive property

denoted by LPO LX



  7.

denoted by LPC LX

I. INTRODUCTION

p  close sets are



Note: The close set in L  bts must be p  close set, and on the contrary generally does not set up.

Since J. C. Kelly introduced the concept of double topological space, many scholars has a great interest about researching dual topology, then introduced L  bitopological spaces on the basis of L  topological space, and makes a long-term study of the separation .Connectivity is an important branch of fuzzy topology, the domestic scholars have studied the variety of connectivity, such as   connectivity[3], connectivity[4], stratified connectedness [5], disconnectedness branches e .c. Due to the concept of connectivity is closely related to geometric closure , many connectivity is also defined based on the definition of different closure concepts.

In this paper,we introduced   P open sets and   P closed sets





Lemma 2.1: Let LX ,  be L  bts , then

  ,    LPC L .

  LPO L

X*

X*

Definition *

2.3:

L ,   X

Let

L  bts

be

,

A L , B  L ，then: X

X*

a  The union of all

p  open sets which contained by

A is called internal of A ’s LE  p  ,record as A ,that is



 



A*  B  LPO LX B  A .

in L  Bitopological spaces on the basis of   P open sets and   P closed sets .In this case ,we defined a new connectivity

b  The union of all

in L  bi-topological spaces which is called   P local-connectivity ,and the study of the connectivity has gotten some good properties.

*

p  close sets which contained by

A is called external of A ’s LE  p  ,record as A is



 



,that



A*  B  LPC LX A  B

II. PRELIMINARY KNOWLEDGE

*

Lemma 2.2: Let LX ,   be L  bts , A, B  LX ,then *

A. L  bitopological spaces

(a) A

Definition 2.1: Let L be a F lattice ,that is a completely distributive lattice with the reverse involution ,let X be a common set and let LX is a set that contains the whole L  fuzzy sets on X ,0 and 1 respectively expressed the

 A  A  A  A .



 ,then A    . (c) If A    LPO L ,then A   . (b) If A    LPC LX



X

L , M L and M  LX  respectively expressed all molecules of L and LX ,we record minimum and maximum in



(d) If A  B ,then A (e)  A  B



 B  , A  B  .

 A  B ,  A  B  A  B  .

(f) The arbitrary intersection of p  closed sets is closed set,and the arbitrary intersection of p  open p sets is p  open set.

FangLi Li, School of Science, Shandong University of Technology, ZiBo, China Hongkui Li, School of Science, Shandong University of Technology, ZiBo, China.

Tao Li, Innovation and entrepreneurship College, Shandong University of Technology, Zibo, China.

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  P Conectedness in L  Bitopological spaces

（f） l A  B   l A   l B  .

C. Figures The Introduction of   p open set And   p close set

D.   p local-connectivity

According to the preparation,we introduce   p open set and   p close set into L  bitopological spaces,and give

1.Definitions about   p local-connectivity Definition 4.1.1:Let LX ,   be

some related properties and theorems 7 about   p open set and   p close set on the basis of p  closed set and p  open set,and give the corresponding proof.



L  bts , A, B  LX ,   L  0 ，if A  B    and A  B     ,then call A and B are   p insular.



Definition 3.1: Let LX ,  be



there is no





l  A  l   A ，then call A as   p open set;If A is









  close set,then about X



  l  A ,so A is a   p close set.

Lemma 3.2:Let L ,   be L  bts , A  L

X



1 p

"" Due to " A" is   1  0, l  A   l   A

is is

Theorem 4.2.1: The local connected space must be

Proof : From the definition of local-connected spaces ,we can know that every neighborhood of A contains a   p connected neighborhood V ,the V is open set .By definition 4.2 in the literature 3 ,we can know that the close

set,so

connected open set must be x  L





(a) LX is a   p local-connected space. (b)Arbitrary   p connected branch of LX ’s arbitrary   p open set is a   p open set.

（a）If A  B ,then l  A  l B  . （b） l  A  B   l  A  l B  .

(c) LX is a connected base.

   l A   l B . （d） l  A  B    l A   l B . 



Theorem 4.2.2: Let LX ,  be L  bts , then the following conditions are equivalent:

Lemma 3.4:Let Lx ,   be L  bts , A, B  LX ,   L  0,then



A contains a ,then LX ,   is

On the contrary, does not necessarily set up.

x  l 0 A is conflict with （*）,so A  A ,that is A is   p close set.



of

 V   p local-connected space according to definition 4.1.4.

A  A ,then there is x  X ,cause  0  Ax   0 ,then A x   A X  ,marked



  p open set of V .As for

X

,if every neighborhood  p connected neighborhood

If



X

  p local-connected space.

"" Clearly established.



L ,  

X

Proof :



connected x is is not

local-connectivity,then call p L ,     p local-connected space.

2 .Basic properties of   p local-connectivity

,if A is a

A is 1  p close set.

（c） l  A  B 

x  LX ,if

X

Lemma 3.3:Let Lx ,  be L  bts , A  L ,If A is a p  close set is the necessary and sufficient condition of





Definition 4.1.4: If every point of

  open set,then A is   p open set.



X

  p local-connectivity 8 .



x

X

Definition 4.1.3: Let L ,  be L  bts , every neighborhood of A contains a   p neighborhood ,then call V x local-connectivity;Otherwise,call p

  L  0, A  L ,when    ,there is l  A    l   A ,due to A  A ,then there is l A



set;Otherwise,call L ,  is   p disconnected space.

X

  close set,then A is   p close set.



is   p Connected

the Maximum element in LX

Lemma 3.1:Let L ,  be L  bts , A  L ,if A is

Proof: If A is



is Connected set in LX ,  .Particularly, when 1X which is

  p close set,then we call A as   p open set . x

X

A and B are A, B  LX ,to make   p insular,and A  B  S , A    , B    ,then call S

l  A  x  X Ax    ,let   L  0,if





Definition 4.1.2:Let LX ,  be L  bts , S  L ,if

L  bts ,   L , A  LX ,record as













Proof :



(a)  (b):





Let U is a arbitrary   p open set of   p connected space,c is arbitrary   p connected branch



（e） l A  B   l A  l B  .

of

31

U

,as for arbitrary x  c U ,then U X  U .And because

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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-9, September 2016 from V  V1  V2  ...  Vn Vi  V i  1,2,..., n  ,and literature 10 ,we can know   p connected nature is finitely

  p local-connected,there is a connected neighborhood V ,and V also is   p connected LX

is

neighborhood of subspace U 9 .

productive property ,so V is a   p connected base of product space LX .Also,from theorem4.2.2,we can know that product space LX  LX 1  LX 2  ....  LX n is also a   p local-connected space.

 (c)：

 is a set family that formed by all   p open X set of L ,then  is a base of L ,also because   p connected branch is   p connected subset,then  is X connected base of L .

Let

Some properties P of topological spaces are called finite integrable properties, if there are n  1 arbitrary topological

X

space LX 1 ， LX 2 ，... LX n have property P ,and contained product space L  L 1  L 2  ....  L n also have property P ,then   p local-connected nature is finitely productive property.

X

 (a)： Let LX has a   p connected base  ,then each member of  are all   p connected sets,as for

x  LX ,make  x  A   x  B,then x   x   ,that

  p connected,we can know   p local-connected by the definition 4.1.3.

is



LX

[1]

is [2]



[3]

space, f : L  X is a continuous open mapping,then we X



space by  X , T  .

 is

[4]

a   p local-connected

[5]

LX is a   p local-connected space, X then we can know there is a   p connected base in L by the definition 4.2.2.As for f is a continuous open Proof :Let

mapping,to

make

 0   f B  B   

[6]

[7]

,then

0

X

Kelly J C. Bitopological Spaces[J]. Proc London Math Soc, 1963(77): 71-89.

Zhang Chunbing,Zheng Chongyou.Separation axiom of L-double fuzzy topological space[J].Journal of Beijing Teachers College (NATURAL SCIENCE EDITION).1990.11(1): 13-17.

Liu Hongping,Meng Guangwu,Hu Xiaonan.The  Connectivity of

L  bitopological spaces.Journal of Shantou University (NATURAL SCIENCE EDITION).2008.05.

Zhao Haixin,Meng Guangwu.The Connectivity of L  bitopological spaces[J].Journal of Jinggangshan University (NATURAL SCIENCE EDITION).2010.11.15.

Xu Hao,Meng Guangwu.The stratified connectedness of L  bitopological spaces[J].Journal of Liaocheng University.2010.06.25.

Wang Xiaoxia,Jiang Jinping.Several classes of weak-connectedness branches of L  bitopological spaces[D].Journal of Yunnan Normal University.2013.07.

Xu Xiaoling.The   p Connectivity of L  fuzzy with L-double

fuzzy topological space.2009.06.

Xiong Jincheng.Point set topology notes[D].Fourth edition.2011,06.

Jiang Pingyuan,Wang Xiaoxia.   local connected space.Jiangxi Science.Volume 26,First period.2008,02.

[10] Zhang Xiangting.Calculation of wind pressure and wind induced vibration of structures[M].Shanghai.Tongji University press, 1985.

B   , f B , B   , f B is   p connected open set

[8] [9]

  p connected open set family of X .As for arbitrary   p open set of X , f is   p connected of X ,so

X

REFERENCES

x

Theorem 4.2.3: Let is a LX 1 ,  1   p local-connected space,  X , T  is a topological can construct that LX 2 ,  2

X

is

and surjection,so:

A f(f Then

0



 A) 

is a

Corresponding Author: Hongkui Li Acknowledgment: This work is supported by Doctoral Fund of Shandong University of Technology of China(4041/415061), Comprehensive budget management system design(9101/215043), Shandong Research project of teaching reform in undergraduate colleges and universities of China(2015Z071).

  f   B    f B  B    1  B1

  p open connected base of X .



By the theorem 4.2.2,we can know LX 2 ,  2

structured by  X , T  is

  p connected space .

 can

We call the nature is topological invariance which is can keep the same nature under   p continuous mapping in topological space.

Theorem 4.2.4: If LX 1 ， LX 2 ， ... LX n are also X X X X   p local-connected space,then L  L 1  L 2  ....  L n is   p local-connected space too.

Proof :Let LX i i  1,2,..., n  is   p connected space,from theorem 4.2.2, we can know that there is a   p connected

base Vi i  1,2,..., n  in

LX i ,then make

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