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P
Preface
Thismonographistheoutcomeofourefforttopresentsomeofour
recentworks,throughanoperatorcalledadherencedominator,inauni-
fiedmode.Thisprojectwasoriginatedbyourobservationsthatmany
significanttopologicalpropertiessuchascompactnessanditsgen-
eralizations,severaltypesofcontinuity,convergenceandseparation
axioms,whichplaymajorrolesintheformerproperties,alldepend
onhowatopologydefinestheclosenessofapointtopointorapoint
toasetorasettoanotherset.Withaset-theoreticalbackground,the
conceptoffiltersbecomeaconvenienttooltostudyconvergence.Also,
adherence,thesetofadherentpointsofafilter,
Ω
,theintersectionof
closureofallmembersof
Ω
,isaclosedset.Sothedefinitionofan
adherencedominatoronatopologicalspace
X
,asafunction
π
fromthe
collectionoffilterbaseson
X
tothefamilyofclosedsubsetsof
X
satisfying
A(Ω)⊂π(Ω)
where
A(Ω)
istheadherenceof
Ω
,wasintroducedby
Joseph[J7]in1980.Josephthenunifiedmanyexistingandnewchar-
acterizationsoftheclassesofminimal-
P
-spacesand
P
-closedspaces,
usingthisoperator.That1980articleprovidedthemotivationofsome
ofourrecentworksandalsoforthismonograph.
Theresultspresentedinthisstudyareorganizedinsixchapters.
InChapter1,theadherencedominatoroperatorisintroducedasacon-
ceptwhichunifiesdifferentclosureoperatorssuchas
θ
-closure,
u
-
closureetc.byintroducingatopologygeneratedbythecomplement
of
π
closedsets,whereaset
A⊆X
is
π
-closedif
πA=A
.Inthiscase
π
istheclosureoperator,andisstatedas
π
closed,whereas,ifasubset
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