More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every ...
Openset
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Jumptonavigation
Jumptosearch
Basicsubsetofatopologicalspace
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Example:Thebluecirclerepresentsthesetofpoints(x,y)satisfyingx2+y2=r2.Thereddiskrepresentsthesetofpoints(x,y)satisfyingx2+y20alwaysbutasεbecomessmallerandsmaller,oneobtainspointsthatapproximatextoahigherandhigherdegreeofaccuracy.Forexample,ifx=0andε=1,thepointswithinεofxarepreciselythepointsoftheinterval(−1,1);thatis,thesetofallrealnumbersbetween−1and1.However,withε=0.5,thepointswithinεofxarepreciselythepointsof(−0.5,0.5).Clearly,thesepointsapproximatextoagreaterdegreeofaccuracythanwhenε=1.
Thepreviousdiscussionshows,forthecasex=0,thatonemayapproximatextohigherandhigherdegreesofaccuracybydefiningεtobesmallerandsmaller.Inparticular,setsoftheform(−ε,ε)giveusalotofinformationaboutpointsclosetox=0.Thus,ratherthanspeakingofaconcreteEuclideanmetric,onemayusesetstodescribepointsclosetox.Thisinnovativeideahasfar-reachingconsequences;inparticular,bydefiningdifferentcollectionsofsetscontaining0(distinctfromthesets(−ε,ε)),onemayfinddifferentresultsregardingthedistancebetween0andotherrealnumbers.Forexample,ifweweretodefineRastheonlysuchsetfor"measuringdistance",allpointsarecloseto0sincethereisonlyonepossibledegreeofaccuracyonemayachieveinapproximating0:beingamemberofR.Thus,wefindthatinsomesense,everyrealnumberisdistance0awayfrom0.Itmayhelpinthiscasetothinkofthemeasureasbeingabinarycondition:allthingsinRareequallycloseto0,whileanyitemthatisnotinRisnotcloseto0.
Ingeneral,onereferstothefamilyofsetscontaining0,usedtoapproximate0,asaneighborhoodbasis;amemberofthisneighborhoodbasisisreferredtoasanopenset.Infact,onemaygeneralizethesenotionstoanarbitraryset(X);ratherthanjusttherealnumbers.Inthiscase,givenapoint(x)ofthatset,onemaydefineacollectionofsets"around"(thatis,containing)x,usedtoapproximatex.Ofcourse,thiscollectionwouldhavetosatisfycertainproperties(knownasaxioms)forotherwisewemaynothaveawell-definedmethodtomeasuredistance.Forexample,everypointinXshouldapproximatextosomedegreeofaccuracy.ThusXshouldbeinthisfamily.Oncewebegintodefine"smaller"setscontainingx,wetendtoapproximatextoagreaterdegreeofaccuracy.Bearingthisinmind,onemaydefinetheremainingaxiomsthatthefamilyofsetsaboutxisrequiredtosatisfy.
Definitions[edit]
Severaldefinitionsaregivenhere,inanincreasingorderoftechnicality.Eachoneisaspecialcaseofthenextone.
Euclideanspace[edit]
Asubset
U
{\displaystyleU}
oftheEuclideann-spaceRnisopenif,foreverypointxin
U
{\displaystyleU}
,thereexistsapositiverealnumberε(dependingonx)suchthatapointinRnbelongsto
U
{\displaystyleU}
assoonasitsEuclideandistancefromxissmallerthanε.[1]Equivalently,asubset
U
{\displaystyleU}
ofRnisopenifeverypointin
U
{\displaystyleU}
isthecenterofanopenballcontainedin
U
.
{\displaystyleU.}
Metricspace[edit]
AsubsetUofametricspace(M,d)iscalledopenif,foranypointxinU,thereexistsarealnumberε>0suchthatanypoint
y
∈
M
{\displaystyley\inM}
satisfyingd(x,y)
