McCluskey A., McMaster B. - Topology Course Lecture Notes.pdf

(255 KB) Pobierz
136944064 UNPDF
TopologyCourseLectureNotes
AislingMcCluskeyandBrianMcMaster
August1997
Chapter1
FundamentalConcepts
Inthestudyofmetricspaces,weobservedthat:
(i)manyoftheconceptscanbedescribedpurelyintermsofopensets,
(ii)open-setdescriptionsaresometimessimplerthanmetricdescriptions,
e.g.continuity,
(iii)manyresultsabouttheseconceptscanbeprovedusing only thebasic
propertiesofopensets(namely,thatboththeemptysetandtheun-
derlyingset X areopen,thattheintersectionofanytwoopensetsis
againopenandthattheunionofarbitrarilymanyopensetsisopen).
Thispromptsthequestion:Howfarwouldwegetifwestartedwithacollec-
tionofsubsetspossessingtheseabove-mentionedpropertiesandproceeded
todefineeverythingintermsofthem?
1.1DescribingTopologicalSpaces
Wenotedabovethatmanyimportantresultsinmetricspacescanbeproved
usingonlythebasicpropertiesofopensetsthat
² theemptysetandunderlyingset X arebothopen,
² theintersectionofanytwoopensetsisopen,and
² unionsofarbitrarilymanyopensetsareopen.
1
Wewillcallanycollectionofsetson X satisfyingthesepropertiesatopology.
Inthefollowingsection,wealsoseektogivealternativewaysofdescribing
thisimportantcollectionofsets.
1.1.1DefiningTopologicalSpaces
Definition1.1 A topologicalspace isapair ( X;T ) consistingofasetX
andafamilyTofsubsetsofXsatisfyingthefollowingconditions:
(T1);2TandX2T
(T2)Tisclosedunderarbitraryunion
(T3)Tisclosedunderfiniteintersection.
Theset X iscalleda space ,theelementsof X arecalled points ofthespace
andthesubsetsof X belongingto T arecalledopeninthespace;thefamily
T ofopensubsetsof X isalsocalledatopologyfor X .
Examples
(i)Anymetricspace( X;d )isatopologicalspacewhere T d ,thetopology
for X inducedbythemetric d ,isdefinedbyagreeingthatGshallbe
declaredasopenwhenevereach x in G iscontainedinanopenball
entirelyin G ,i.e.
;½GµX isopenin( X;T d ) ,
8x2G;9r x > 0suchthat x2B r x ( x ) µG:
(ii)Thefollowingisaspecialcaseof(i),above.Let R bethesetofreal
numbersandlet I betheusual(metric)topologydefinedbyagreeing
that
;½GµX isopenin( R;I )(alternatively, I -open) ,
8x2G;9r x > 0suchthat( x¡r x ;x + r x ) ½G:
(iii)Define T 0 = f;;Xg foranysetX—knownasthe trivial or anti-discrete topology.
(iv)Define D = fGµX : GµXg —knownasthe discrete topology.
2
(v)Foranynon-emptyset X ,thefamily C = fGµX : G = ; or XnG is
finite g isatopologyfor X calledthe cofinite topology.
(vi)Foranynon-emptyset X ,thefamily L = fGµX : G = ; or XnG is
countable g isatopologyfor X calledthe cocountable topology.
1.1.2Neighbourhoods
Occasionally,argumentscanbesimplifiedwhenthesetsinvolvedarenot
“over-described”.Inparticular,itissometimessu±cestousesetswhich
containopensetsbutarenotnecessarilyopen.Wecallsuchsetsneighbor-
hoods.
Definition1.2 Givenatopologicalspace ( X;T ) withx2X,thenNµX
issaidtobea(T)- neighbourhood ofx,9opensetGwithx2GµN.
Itfollowsthenthataset UµX isopeni®forevery x2U ,thereexistsa
neighbourhood N x of x containedin U .(Checkthis!)
Lemma1.1 Let ( X;T ) beatopologicalspaceand,foreachx2X,letN ( x )
bethefamilyofneighbourhoodsofx.Then
(i)U2N ( x ) )x2U:
(ii)N ( x ) isclosedunderfiniteintersections.
(iii)U2N ( x ) andUµV)V2N ( x ) :
(iv)U2N ( x ) )9W2N ( x ) suchthatWµUandW2N ( y ) foreach
y2W:
ProofExercise!
Examples
(i)Let x2X ,anddefine T x = f;;fxg;Xg .Then T x isatopologyfor X
and VµX isaneighbourhoodof x i® x2V .However,theonlynhd
of y2X where y6 = x is X itself
3
(ii)Let x2X anddefineatopology I ( x )for X asfollows:
I ( x )= fGµX : x2Gg[f;g:
Noteherethat every nhdofapointin X isopen.
(iii)Let x2X anddefineatopology E ( x )for X asfollows:
E ( x )= fGµX : x62Gg[fXg:
Noteherethat fyg isopenforevery y6 = x in X ,that fx;yg is not
open,isnotanhdof x yet is anhdof y .
Infact,theonlynhdof x is X .
1.1.3BasesandSubbases
Itoftenhappensthattheopensetsofaspacecanbeverycomplicatedand
yet theycanallbedescribedusingaselectionoffairlysimplespecialones.
Whenthishappens,thesetofsimpleopensetsiscalledabaseorsubbase
(dependingonhowthedescriptionistobedone).Inaddition,itisfortunate
thatmanytopologicalconceptscanbecharacterizedintermsofthesesimpler
baseorsubbaseelements.
Definition1.3 Let ( X;T ) beatopologicalspace.AfamilyBµTiscalled
a basefor( X;T ) ifandonlyifeverynon-emptyopensubsetofXcanbe
representedasaunionofasubfamilyofB.
Itiseasilyverifiedthat BµT isabasefor( X;T )ifandonlyifwhenever
x2G2T;9B2B suchthat x2BµG .
Clearly,atopologicalspacecanhavemanybases.
Lemma1.2 IfBisafamilyofsubsetsofasetXsuchthat
(B1)foranyB 1 ;B 2 2Bandeverypointx2B 1 \B 2 ,thereexistsB 3 2B
withx2B 3 µB 1 \B 2 ,and
(B2)foreveryx2X,thereexistsB2Bsuchthatx2B,
thenBisabaseforauniquetopologyonX.
Conversely,anybaseBforatopologicalspace ( X;T ) satisfies(B1)and(B2).
Proof(Exercise!)
Definition1.4 Let ( X;T ) beatopologicalspace.AfamilySµTiscalled
a subbasefor( X;T ) ifandonlyifthefamilyofallfiniteintersections
\ k i =1 U i ,whereU i 2Sfori =1 ; 2 ;:::;kisabasefor ( X;T ) .
4

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin