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7
iftheredoesnotexista
P
-topology
τ!
on
X
weakerthan
τ
.Aspace
(X
,
τ)
is
P
-closedifitisclosedinany
P
-spaceinwhichitcanbeem-
bedded.Inthesurveyarticle[BPS],thereisacatalogueofresultson
minimal-
P
-spacesand
P
-closedspacesanditgivesdetaliedrelation-
shipsbetweentheseclassesofspaces.Itiswellknownthatacompact
spaceisminimal-
P
aswellas
P
-closedforthetopologicalproperties
weconsiderandhencetheseclassesofspacesgivegeneralizations
ofcompactspaces.So,itishelpfultoputthosepropertiesinauni-
fiedframework.Alongwithpropertiesoftheseclassesofspaces,the
generalframeworkalsocouldprovideproperttiesofothergenerali-
zationsofcompactnesssuchasnearlycompactspacesofSingaland
Mathur[SM].Thischapterisdividedintofivesections.InSection
4.1,introduction,notations,definitionsandconceptswhichareused
intherestofthechapteraregiven.InSections4.2and4.3
P
-closed
spacesandminimal-
P
-spacesaregiven,Section4.4dealswithfirst
countablecompactspacesandminimal-
P
-spacesandSection4.5gives
applicationsoftheseresults.
Chapter5isastudyofcompactness.Herewefocusrecentre-
sultsoncompactnessincludedin[EJN],[EJKN1],[EJKN2],[EJKN3]
and[JN1].Inthesearticles,weaddressthefollowingquestionsstated
in[BPS]:Isaregularspaceinwhicheveryclosedsubsetisregular-
closedcompact?andisaUrysohn-spaceinwhicheveryclosedsub-
setisUrysohn-closedcompact?Wefirstaddressedthesequestions
in[JN1].ToanswerthequestionforHausdorff-closedspacesinthe
affirmative,M.H.Stone[Sto]usedBooleanringsandM.Kat
etov[Ka]
ˇ
usedtopologicalmethods.In[EJN],allthreequestionsareanswered
affirmativelyusingfilters.Ageneralapproachisadaptedinthese
articlestoprovecompactnessusingtheresultthataspaceiscom-
pactifandonlyif,everyultrafilterconverges.Theseproblemswere
alsoaddressedin[Ste].In[EJN],somegapsin[JN1]and[EJKN1]
whichwerepointedoutin[Ste]areaddressed.Inthischapter,com-
pactnessisstudiedthroughsubsetswhichsatisfypropertiesweaker
thancompactness,suchas
P
-closedspaces,fordifferentproperties
P
,
paracompactness,metacompactness,differentconceptsof
P(i)
spaces
andlocally
P(i)
spaces.Characterizingparacompactnessandmeta-
P
