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6
P
is
π
-closedasasubspace,itsatisfiesthepropertyofaspacebeing
π
-closedspace,asdefinedinChapter3andstudiedindetailinChap-
ter4,usingtheadherencedominatoroperator.Detailedrelationships
betweendifferentclosureoperatorsandcorrespondingadherencesets
ofafilterbasearepresentedbeforeintroducingtheadherencedomi-
natoroperator.Thisisdoneinordertoguidethereadertowardsthe
unifyingnatureoftheclosureoperator.Inthatprocess,propertiesof
afilterbase,filterandultrafiltersaregivenheresincethoseproperties
willbeusedthroughoutintherestofthismornograph.
Chapter2presentscontinuousfunctionsandseparationaxioms.
Theseparationaxiomsaredefinedusingtheadherencedominator
π
,
usinganalogouscharacterizationsof
T1
-spaces,Hausdorffspacesand
Urysohnspaces,inthepresenceoftheclosureoperator,
θ
-closureoper-
atorand
u
-closureoperatorrespectively,asmodels.Forcontinuity,the
characterizationthatafunction
f:X→Y
iscontinuousifandonlyif
f(clA)⊆clf(A)
givesthefoundationfordefiningafunction
f:X→Y
is
π
-continuousif
f(πΩ)⊂πf(Ω)
foreveryfilterbase
Ω
.Beforepresent-
ingthisformofconituity,severalgeneralizationsofcontinuity,their
characterizationsandinterrelationshipsarepresented.Mostofthe
resultsgiveninthisChapterarefromourresultsin[JN4]and[JKN1].
Chapter3dealswithconnectedness.Webelievethattheconceptof
connectednesscanbeabstractedfrompropertiesembeddedintheset
ofrealnumbersystem.ClayandJoseph[CJ]studiedconnectedness
ofasetusing
θ
-closureoperator.Themodelofa
θ
-connectedset
ofClayandJoseph[CJ]isadaptedin[JN4]tostudyconnectedness
usingtheadherencedominatoroperator.Inthischapter,thestudy
ofconnectednessusingadherencedominator,wefollowthestudy
in[JN4].Asubset
K
iscalled
π
-connectedrelativeto
X
if
K=∅
or
K/=P∪Q
,with
P∩πQ=πP∩Q=∅
,where
π
isanadherence
dominatoron
X
.Thatistosay
K=∅
or
K
hasno
π
-partition(or
π
-
separation).
Chapter4presentsminimal-
P
-spacesand
P
-closedspacesforvari-
oustopologicalproperties
P
throughtheoperator
π
,theadherence
dominator.Thematerialsofthischapterarefrom[J7],thearticle
whichmotivatedthisstudy.A
P
-space
(X
,
τ)
issaidtobeminimal-
P
