the subject of ideals in topological space has been studied by kuratowski   and vaidyanathaswamy . an ideal  on a topological space ( x ,  ) is a nonempty collection of subsets of x which satisfies ( i ) a   and b  a implies b   and ( ii ) a   and b   implies a  b  . given a topological space ( x ,  ) with an ideal  on x and if (x ) is the set of all subsets of x , a set operator (  ) : (x )  (x ) , called a local function   of a with respect to  and  , is defined as follows : for a  x , a( ,  ) = { x  xu   } , for every { u  (x ) } where (x ) = { u  x  u}. a kuratowski closure operator cl ( ) for a topology (x ,  ) , called the -topology , finer than  is defined by cl(a ) = a  a( ,  ) ; when there is no chance for confusion , we will simply write a for a( ,  ) and  for ( ,  ) . if  is an ideal on x , then ( x ,  ,  ) a of an ideal space ( x ,  ,  ) is said to be -open   if a  int(a ) . a subset a of an ideal space ( x ,  ,  ) is said to be pre--open   if a  int(cl(a ) ) . the family of all pre--open sets in ( x ,  ,  ) is denoted by po(x ,  ) or simply po(x ) . clearly   po(x ) . the largest pre--open set contained in a , denoted by pint(a ) , is called the pre--interior of a. the smallest pre--closed set containing a , denoted by pcl(a ) , is called the pre--closure of a. a subset a of an ideal space ( x ,  ,  ) is said to be --open   if a  int(cl(int( ) ) ) . the family of all --open sets is a topology finer than . we will denote the --interior subset of a of x by int(a ) . a subset a of an ideal space ( x ,  ,  ) is -dense   if cl(a ) = x. a space ( x ,  ,  ) is -submaximal   if every -dense subset of x is open . let ( x ,  ,  ) be an ideal space and let a be a subset of x. then pint(a ) = aint(cl(a ) ) . let ( x ,  ,  ) be an ideal space and let a be a subset of x. then pint(a ) = aint(cl(a ) ) . a pre--open set a of a space ( x ,  ,  ) is said to be pre--regular pre--open if a = pint(pcl(a ) ) . the complement of a pre--regular pre--open set is called pre--regular pre--closed set , equivalently a = pcl(pint(a ) ) . a subset a of a space ( x ,  ,  ) with an ideal  is said to be pre--regular if it is pre--open and pre--closed . a is pre--open which implies a = pint(a ) = pint(pcl(a ) ) . hence a is pre--regular pre--open . but consider the ideal space ( x ,  ,  ) where x = { a , b , c } ,  = {  , { a } , { b } , { a , b } , x } , and  = {  , { a}}. here { a } is pre--regular pre--open but not pre--closed . consider the ideal space ( x ,  ,  ) where x = { a , b , c } ,  = {  , { a } , { b } , { a , b } , x } , and  = {  , { a}}. here { a } is pre--regular pre--open but not pre--closed . moreover , the intersection of two pre--regular pre--open sets is not pre--regular pre--open in general as example 4 shows . consider the ideal space ( x ,  ,  ) where x = { a , b , c } ,  = {  , { b , c } , x } , and  = {  , { a}}. here po(x ) = {  , { b } , { c } , { a , b } , { a , c } , { b , c } , x}. thus , a = { a , b } , b = { a , c } are both pre--regular pre--open . consider the ideal space ( x ,  ,  ) where x = { a , b , c } ,  = {  , { b , c } , x } , and  = {  , { a}}. here po(x ) = {  , { b } , { c } , { a , b } , { a , c } , { b , c } , x}. thus , a = { a , b } , b = { a , c } are both pre--regular pre--open . consider the space ( x ,  ) with an ideal  as in example 4 . here consider the space x = { a , b , c } and  = {  , { a } , { b } , { a , b } , x } ,  = {  , { a}}. here { a } is pre--regular pre--open but not -open . observe that every pre--regular pre--open set is pre--open but the converse need not be true . here let ( x ,  ,  ) be an ideal space and let a , b be any subsets of x. then the following hold.(a)if a  b , then pint(pcl(a ) )  pint(pcl(b)).(b)if a  pio(x ) , then a  pint(pcl(a)).(c)for every a  po(x ) , pint(pcl(pint(pcl(a ) ) ) ) = pint(pcl(a)).(d)if a and b are disjoint pre--open sets , then pint(pcl(a ) ) and pint(pcl(b ) ) are disjoint.(e)if a is a pre--regular pre--open , then pcl(x  a ) is pre--regular pre--closed.(f)if a is pre--regular pre--open , then pint(a ) is pre--regular pre--open . let ( x ,  ,  ) be an ideal space and let a , b be any subsets of x. then the following hold.(a)if a  b , then pint(pcl(a ) )  pint(pcl(b)).(b)if a  pio(x ) , then a  pint(pcl(a)).(c)for every a  po(x ) , pint(pcl(pint(pcl(a ) ) ) ) = pint(pcl(a)).(d)if a and b are disjoint pre--open sets , then pint(pcl(a ) ) and pint(pcl(b ) ) are disjoint.(e)if a is a pre--regular pre--open , then pcl(x  a ) is pre--regular pre--closed.(f)if a is pre--regular pre--open , then pint(a ) is pre--regular pre--open . if a  b , then pint(pcl(a ) )  pint(pcl(b ) ) . if a  pio(x ) , then a  pint(pcl(a ) ) . for every a  po(x ) , pint(pcl(pint(pcl(a ) ) ) ) = pint(pcl(a ) ) . if a and b are disjoint pre--open sets , then pint(pcl(a ) ) and pint(pcl(b ) ) are disjoint . if a is a pre--regular pre--open , then pcl(x  a ) is pre--regular pre--closed . proof 
 ( a)suppose a  bpcl(a )  pcl(b ) . therefore pint(pcl(a ) )  pint(pcl(b)).(b)suppose that a  po(x , )a = pint(a )  pint(pcl(a)).(c)it is obvious that pint(pcl(a ) )  po(x ,  ) , so by ( b ) we have pint(pcl(a))pint(pcl(pint(pcl(a ) ) ) ) . on the other hand pint(pcl(a ) )  pcl(a ) which implies pcl(pint(pcl(a)))pcl(pcl(a ) ) = pcl(a ) . therefore , pint(pcl(pint(pcl(a))))pint(pcl(a ) ) . hence pint(pcl(pint(pcl(a ) ) ) ) = pint(pcl(a)).(d)since a and b are disjoint pre--open sets , we have ab =  which implies apcl(b ) = apint(pcl(b ) ) = . since pint(pcl(b ) ) is pre--open , pcl(a)pint(pcl(b ) ) = . hencepint(pcl(a))pint(pcl(b ) ) = .(e)given that a is pre--regular pre--open , a = pint(pcl(a ) ) implies x  a = x  pint(pcl(a ) ) = pcl(pint(x  a ) ) . therefore , pcl(x  a ) = pcl(pint(pcl(x    a ) ) ) . hence pcl(x  a ) is pre--regular pre--closed.(f)by ( e ) if a is pre--regular pre--open , then pcl(x  a ) is pre--regular pre--closed . hence x  ( pcl(x  a ) ) is pre--regular pre--open that implies pint(a ) is pre--regular pre--open . ( a)suppose a  bpcl(a )  pcl(b ) . therefore pint(pcl(a ) )  pint(pcl(b)).(b)suppose that a  po(x , )a = pint(a )  pint(pcl(a)).(c)it is obvious that pint(pcl(a ) )  po(x ,  ) , so by ( b ) we have pint(pcl(a))pint(pcl(pint(pcl(a ) ) ) ) . on the other hand pint(pcl(a ) )  pcl(a ) which implies pcl(pint(pcl(a)))pcl(pcl(a ) ) = pcl(a ) . therefore , pint(pcl(pint(pcl(a))))pint(pcl(a ) ) . hence pint(pcl(pint(pcl(a ) ) ) ) = pint(pcl(a)).(d)since a and b are disjoint pre--open sets , we have ab =  which implies apcl(b ) = apint(pcl(b ) ) = . since pint(pcl(b ) ) is pre--open , pcl(a)pint(pcl(b ) ) = . hencepint(pcl(a))pint(pcl(b ) ) = .(e)given that a is pre--regular pre--open , a = pint(pcl(a ) ) implies x  a = x  pint(pcl(a ) ) = pcl(pint(x  a ) ) . therefore , pcl(x  a ) = pcl(pint(pcl(x    a ) ) ) . hence pcl(x  a ) is pre--regular pre--closed.(f)by ( e ) if a is pre--regular pre--open , then pcl(x  a ) is pre--regular pre--closed . hence x  ( pcl(x  a ) ) is pre--regular pre--open that implies pint(a ) is pre--regular pre--open . suppose a  bpcl(a )  pcl(b ) . therefore pint(pcl(a ) )  pint(pcl(b ) ) . it is obvious that pint(pcl(a ) )  po(x ,  ) , so by ( b ) we have pint(pcl(a))pint(pcl(pint(pcl(a ) ) ) ) . on the other hand pint(pcl(a ) )  pcl(a ) which implies pcl(pint(pcl(a)))pcl(pcl(a ) ) = pcl(a ) . therefore , pint(pcl(pint(pcl(a))))pint(pcl(a ) ) . hence pint(pcl(pint(pcl(a ) ) ) ) = pint(pcl(a ) ) . since a and b are disjoint pre--open sets , we have ab =  which implies apcl(b ) = apint(pcl(b ) ) = . since pint(pcl(b ) ) is pre--open , pcl(a)pint(pcl(b ) ) = . hencepint(pcl(a))pint(pcl(b ) ) = . given that a is pre--regular pre--open , a = pint(pcl(a ) ) implies x  a = x  pint(pcl(a ) ) = pcl(pint(x  a ) ) . therefore , pcl(x  a ) = pcl(pint(pcl(x    a ) ) ) . hence pcl(x  a ) is pre--regular pre--closed . by ( e ) if a is pre--regular pre--open , then pcl(x  a ) is pre--regular pre--closed . hence x  ( pcl(x  a ) ) lemma 6 . for an ideal topological space ( x ,  ,  ) the following are equivalent . for an ideal topological space ( x ,  ,  ) the following are equivalent . proof(a )  ( b ) : let a be a -dense set that implies cl(a ) = x which implies int(cl(a ) ) = x , so that a  int(cl(a ) ) = x. by ( a ) every pre--open set is open and hence a is open.(b )  ( a ) : let b be a pre--open subset of x , so that b  int(cl(b ) ) = g , say . b ) = cl(x  g )  cl(b ) = ( x  g )  cl(b ) = x , and thus ( x  g )  b is -dense in x. thus ( x  g )  b is open . now b = ( ( x  g )  b)g is the intersection of two open sets , so that b is open . 
  ( a ) ( b ) : let a be a -dense set that implies cl(a ) = x which implies int(cl(a ) ) = x , so that a  int(cl(a ) ) = x. by ( a ) every pre--open set is open and hence a is open . ( b )  ( a ) : let b be a pre--open subset of x , so that b  int(cl(b ) ) = g , say b ) = cl(x  g )  cl(b ) = ( x  g )  cl(b ) = x , and thus ( x  g )  b is -dense in x. thus ( x  g )  b is open . now b = ( ( x  g )  b)g is the intersection of two open sets , so that b is open . if a space ( x ,  ) with an ideal  is -submaximal , then any finite intersection of pre--open set is pre--open . if a space ( x ,  ) with an ideal  is -submaximal , then any finite intersection of pre--open set is pre--open . prooffrom lemma 6 , every pre--open set is open and hence a finite intersection of pre--open set is pre--open . 
  from lemma 6 , every pre--open set is open and hence a finite intersection of pre--open set is pre--open . if a space ( x ,  ) with an ideal  is -submaximal , then any finite intersection of pre--regular p--open set is pre--regular pre--open . if a space ( x ,  ) with an ideal  is -submaximal , then any finite intersection of pre--regular p--open set is pre--regular pre--open . n } be a finite family of pre--regular pre--open sets . since the space x is -submaximal , then by lemma 6 , {oii = 1,2 , therefore , {oii = 1,2 ,  , n}pint(pcl(oi ) ) . also , for each i = 1,2 ,  , n , oi  oi which implies pint(pcl(oi))pint(pcl(oi ) ) . also , each oi is pre--regular pre--open that implies oi = pint(pcl(oi ) ) which implies pint(pcl(oi))oi and so oi = pint(pcl(oi ) ) . n } be a finite family of pre--regular pre--open sets . since the space x is -submaximal , then by lemma 6 , {oii = 1,2 ,  , therefore , {oii = 1,2 ,  , n}pint(pcl(oi ) ) . also , for each i = 1,2 , also , each oi is pre--regular pre--open that implies oi = pint(pcl(oi ) ) which implies pint(pcl(oi))oi and so oi = pint(pcl(oi ) ) . it should be noted that an arbitrary union of pre--regular pre--open set is pre--regular pre--open . but the intersection of two pre--regular pre--closed sets fails to be pre--regular pre--closed as shown by example 9 . consider the ideal space ( x ,  ,  ) as in example 3 . clearly , { a , c } , { a , b } are pre--regular pre--closed but their intersection is not pre--regular pre--closed . consider the ideal space ( x ,  ,  ) as in example 3 . clearly , { a , c } , { a , b } are pre--regular pre--closed but their intersection is not pre--regular pre--closed . the following hold for a subset a of a space ( x ,  , ).(a)if a is pre--closed , then pint(a ) is pre--regular pre--open.(b)if a = pint(a ) , then pcl(a ) is pre--regular pre--closed.(c)if a and b are pre--regular pre--closed sets , then a  b if and only if pint(a )  pint(b).(d)if a and b are pre--regular pre--open sets , then a  b if and only if pcl(a )  pcl(b ) . the following hold for a subset a of a space ( x ,  , ).(a)if a is pre--closed , then pint(a ) is pre--regular pre--open.(b)if a = pint(a ) , then pcl(a ) is pre--regular pre--closed.(c)if a and b are pre--regular pre--closed sets , then a  b if and only if pint(a )  pint(b).(d)if a and b are pre--regular pre--open sets , then a  b if and only if pcl(a )  pcl(b ) . if a is pre--closed , then pint(a ) is pre--regular pre--open . if a = pint(a ) , then pcl(a ) is pre--regular pre--closed . if a and b are pre--regular pre--closed sets , then a  b if and only if pint(a )  pint(b ) . if a and b are pre--regular pre--open sets , then a  b if and only if pcl(a )  pcl(b ) . proof(a ) since a is pre--closed , a = pcl(a).now , pint(pcl(pint(a ) ) ) = pint(pint(a ) ) = pint(a ) . hence pint(a ) is pre--regular pre--open.(b ) now pcl(pint(pcl(a ) ) ) = pcl(pcl(a ) ) = pcl(a ) . hence pcl(a ) is pre--regular pre--closed.(c ) given that a and b are pre--regular pre--closed sets , therefore , a = pcl(pint(a ) ) and b = pcl(pint(b ) ) . clearly , pint(a )  pint(b ) if a  b.conversely , pint(a )  pint(b ) . now a = pcl(pint(a ) )  pcl(pint(b ) )  b. hence a  b.(d ) given that a and b are pre--regular pre--open , therefore , a = pint(pcl(a ) ) and b = pint(pcl(b ) ) . therefore , pcl(a )  pcl(b).conversely , pcl(a )  pcl(b ) . now a = pint(pcl(a ) )  pint(pcl(b ) )  b. 
  ( a ) since a is pre--closed , a = pcl(a ) . ( b ) now pcl(pint(pcl(a ) ) ) = pcl(pcl(a ) ) = pcl(a ) . hence pcl(a ) is pre--regular pre--closed . ( c ) given that a and b are pre--regular pre--closed sets , therefore , a = pcl(pint(a ) ) and b = pcl(pint(b ) ) . clearly , pint(a )  pint(b ) if a  b. conversely , pint(a )  pint(b ) . now a = pcl(pint(a ) )  pcl(pint(b ) )  b. hence a  b. ( d ) given that a and b are pre--regular pre--open , therefore , a = pint(pcl(a ) ) and b = pint(pcl(b ) ) . now a = pint(pcl(a ) )  pint(pcl(b ) )  b. a subset a of an ideal topological space ( x ,  ,  ) is said to be -rare if it has no interior points in . 
 theorem 11 . then the following hold.(a)the empty set is the only subset which is nowhere dense and pre--regular pre--open.(b)if a is pre--regular pre--closed , then every -rare set is pre--open then the following hold.(a)the empty set is the only subset which is nowhere dense and pre--regular pre--open.(b)if a is pre--regular pre--closed , then every -rare set is pre--open . the empty set is the only subset which is nowhere dense and pre--regular pre--open . if a is pre--regular pre--closed , then every -rare set is pre--open . then a = pint(pcl(a ) ) = pcl(a)int(cl(pcl(a ) ) ) , by lemma 1 . therefore , a  pcl(a)int(cl(cl(a)))pcl(a)int(cl(a ) ) = pcl(a) = .(b ) suppose a is pre--regular pre--closed . then a = pcl(pint(a ) ) = pint(a)cl(int(pint(a)))pint(a )  cl(int(a ) ) = pint(a )   = pint(a ) . then a = pint(pcl(a ) ) = pcl(a)int(cl(pcl(a ) ) ) , by lemma 1 . therefore , a  pcl(a)int(cl(cl(a)))pcl(a)int(cl(a ) ) = pcl(a) = . ( b ) suppose a is pre--regular pre--closed . then a = pcl(pint(a ) ) = pint(a)cl(int(pint(a)))pint(a )  cl(int(a ) ) = pint(a )   = pint(a ) . hence a is pre--open . an ideal space ( x ,  ,  ) is called extremally pre--disconnected if the pre--closure of every pre--open set is pre--open . theorem 12 . for a topological space ( x ,  ,  ) the following are equivalent.(a)(x ,  ,  ) is extremally pre--disconnected.(b)every pre--regular pre--open subset is pre--regular . 
  for a topological space ( x ,  ,  ) the following are equivalent.(a)(x ,  ,  ) is extremally proof(a )  ( b ) : assume ( x ,  ,  ) is extremally pre--disconnected . hence a is pre--closed which implies a is pre--regular.(b )  ( a ) : suppose a is pre--open . then pcl(a ) is pre--regular pre--closed which implies x  pcl(a ) is pre--regular pre--open . ( a )  ( b ) : assume ( x ,  ,  ) is extremally pre--disconnected . then pcl(a ) is pre--regular pre--closed which implies x  pcl(a ) is pre--regular pre--open . let ( x ,  ,  ) be an extremally pre--disconnected space and a  x. then the following are equivalent:(a)a is pre--regular,(b)a = pcl(pint(a)),(c)x  a is pre--regular pre--open,(d)a is pre--regular pre--open . let ( x ,  ,  ) be an extremally pre--disconnected space and a  x. then the following are equivalent:(a)a is pre--regular,(b)a = pcl(pint(a)),(c)x  a is pre--regular pre--open,(d)a is pre--regular pre--open . then a is pre--open and pre--closed and so a = pint(a ) and a = pcl(a ) . hence a = pcl(pint(a)).(b )  ( c ) : let a = pcl(pint(a ) ) . then x  a = x  pcl(pint(a ) ) = pint(pcl(a ) ) so x  a is pre--regular pre--open.(c )  ( d ) is clear.(d )  ( a ) follows from theorem 12 . then a is pre--open and pre--closed and so a = pint(a ) and a = pcl(a ) . then x  a = x  pcl(pint(a ) ) = pint(pcl(a ) ) so x  a is pre--regular pre--open . ( d )  ( a ) follows from theorem 12 . an ideal space ( x ,  ,  ) is called locally pre--indiscrete if every pre--open subset of x is pre--closed ( or ) if every pre--closed subset of x is pre--open . then the following are equivalent.(a)(x ,  ,  ) is locally pre--indiscrete.(b)every pre--open subset is pre--regular.(c)every pre--open subset is pre--regular pre--open.(d)pint(pcl({x } ) )   , for every x  x.(e)the empty set is the only nowhere dense subset of x. 
  let ( x ,  ,  ) be an ideal space . then the following are equivalent.(a)(x ,  ,  ) is locally pre--indiscrete.(b)every pre--open subset is pre--regular.(c)every pre--open subset is pre--regular pre--open.(d)pint(pcl({x } ) )   , for every x  x.(e)the empty set is the only nowhere dense subset of x. ( x ,  ,  ) is locally pre--indiscrete . pint(pcl({x } ) )   , for every x  x. the empty set is the only nowhere dense subset of x. 
 proof(a )  ( b ) : assume that ( x ,  ,  ) is locally pre--indiscrete . hence a is pre--regular.(b )  ( c ) : if a is pre--open , then a = pint(a ) . also by hypothesis , hence a is pre--regular pre--open.(c )  ( d ) : since { x } is preopen , { x } is pre--open . by ( c ) , { x } is a pre--regular pre--open set . therefore , { x } = pint(pcl({x})).(d )  ( e ) : by theorem 11 , in any space , the empty set is the only subset which is nowhere dense and pre--regular pre--open.(e )  ( a ) : suppose that a is a pre--closed set . now int(cl(a pint(a ) ) ) = int(cl(a  ( aint(cl(a ) ) ) ) ) = int(cl(a(x  a)(int(cl(x  a ) ) ) ) ) = int(cl(a(x  a)(a  int(cl(a ) ) ) ) ) = int(cl(a  int(cl(a))))int(cl(a )  int(cl(a ) ) ) = int(cl(a )  cl(int(cl(a))))int(cl(a )    ( int(cl(a ) ) ) ) = . therefore a  pint(a ) is nowhere dense which implies a = pint(a ) , and so a is pre--open . hence is a locally pre--indiscrete . ( a )  ( b ) : assume that ( x ,  ,  ) is locally pre--indiscrete . let a be a pre--open subset of x. by hypothesis , a is pre--closed . hence a is pre--regular . ( b )  ( c ) : if a is pre--open , then a = pint(a ) . also by hypothesis , ( c )  ( d ) : since { x } is preopen , { x } is pre--open . by ( c ) , { x } is a pre--regular pre--open set . ( d )  ( e ) : by theorem 11 , in any space , the empty set is the only subset which is nowhere dense and pre--regular pre--open . ( e )  ( a ) : suppose that a is a pre--closed set . now int(cl(a pint(a ) ) ) = int(cl(a  ( aint(cl(a ) ) ) ) ) = int(cl(a(x  a)(int(cl(x  a ) ) ) ) ) = int(cl(a(x  a)(a  int(cl(a ) ) ) ) ) = int(cl(a  int(cl(a))))int(cl(a )  int(cl(a ) ) ) = int(cl(a )  cl(int(cl(a))))int(cl(a )    ( int(cl(a ) ) ) ) = . therefore a  pint(a ) is nowhere dense which implies a = pint(a ) , and so a is pre--open . hence is a locally pre--indiscrete . an ideal space ( x ,  ,  ) is said to be pr - door if every subset of x is either pre--regular pre--open or pre--regular pre--closed . let ( x ,  ,  ) be a pr - door space ; then every pre--open set in the space is pre--regular pre--open . let ( x ,  ,  ) be a pr - door space ; then every pre--open set in the space is pre--regular pre--open . prooflet a be a pre--open subset of x. since x is pr - door , a is pre--regular pre--closed and so a = pint(pcl(a ) ) which implies that pinta = pint(pint(pcl(a ) ) ) . let a be a pre--open subset of x. since x is pr - door , a is pre--regular pre--closed and so a = pint(pcl(a ) ) which implies that pinta = pint(pint(pcl(a ) ) ) . a subset a of x is both --open and --closed ; then a is a pre--regular pre--open set a subset a of x is both --open and --closed ; then a is a pre--regular pre--open set . prooflet a be an --open and --closed set . then a is a pre--open and pre--closed set and hence a is a pre--regular pre--open set . 
  let a be an --open and --closed set . then a is a pre--open and pre--closed set and hence a is a pre--regular pre--open set . let ( x ,  ,  ) be an ideal space . a subset a of x is --open and pre--regular pre--open ; then a = int(cl(int(a ) ) ) . a subset a of x is --open and pre--regular pre--open ; then a = int(cl(int(a ) ) ) . then a  int(cl(int(a ) ) ) . and a is pre--regular pre--open which implies a = pint(pcl(a))pint(cl(int(pcl(a))))pint(cl(int(a)))int(cl(int(a ) ) ) . then a  int(cl(int(a ) ) ) . and a is pre--regular pre--open which implies a = pint(pcl(a))pint(cl(int(pcl(a))))pint(cl(int(a)))int(cl(int(a ) ) ) .
we deal with the new class of pre--regular pre--open sets in which the notion of pre--open set is involved . 
 we characterize these sets and study some of their fundamental properties . 
 we also present other notions called extremally pre--disconnectedness , locally pre--indiscreetness , and pre--regular sets by utilizing the notion of pre--open and pre--closed sets by which we obtain some equivalence relation for pre--regular pre--open sets .