Mis vahe on avatud ja suletud komplektil?


Vastus 1:

Suletud komplektid on komplektid, kus nende komplekt (komplekt kõike, mida komplektis pole) on avatud.

GivenauniversalsetXaset[math]S[/math]isclosedif[math]XS[/math]isopen.Given a universal set X a set [math]S[/math] is closed if [math]X \smallsetminus S[/math] is open.

WhenwehavethesetsthatareconsideredopeninXwealsohaveanotiononwhichsetsareclosed.Beawarethatsetsarentlikedoors.Theycanbeneitheropennorclosed,orbothopenandclosed.When we have the sets that are considered open in X we also have a notion on which sets are closed. Be aware that sets aren’t like doors. They can be neither open nor closed, or both open and closed.

asetSisclosedif[math]XS[/math]isopen. a set S is closed if [math]X \smallsetminus S[/math] is open.

Ontherealnumberlinewiththestandardtopologyopensetsarethosethatdonotincludeanyoftheirboundary.Everypointxinanopenset[math]U[/math]canbenicelyputintoanopeninterval[math]x(a,b)U[/math].On the real number line with the standard topology open sets are those that do not include any of their boundary. Every point x in an open set [math]U[/math] can be nicely put into an open interval [math]x \in (a,b) \subset U[/math].

So(4,7)istheintervalofallrealnumbersstrictlybetween[math]4[/math]and[math]7[/math]andisanopenset.Sois[math](4,2)(1,5)(7,)[/math].So (4,7) is the interval of all real numbers strictly between [math]4[/math] and [math]7[/math] and is an open set. So is [math](-4,2) \cup (1,5) \cup (7,\infty)[/math].

Similarlysetssuchas[0,1],[4,3][8,11],{2},{1,2,3}areclosed.Similarly sets such as [0,1], [-4,3] \cup [8,11], \{2\}, \{1,2,3\} are closed.

Avatud komplektid ei hõlma nende piiri, samas kui suletud komplektid seda teevad.

Pärisnumbri real koos standardse topoloogiaga on avatud komplektid need, mis ei hõlma ühtegi nende piiri. Iga punkt

Uopen[math]xU  r:Br(x)U[/math]U open [math]\Leftrightarrow \forall x \in U \; \exists r : B_{r}(x) \subset U[/math]

inanopensetUcanbenicelyputintoanopeninterval[math]x(a,b)U[/math]. in an open set U can be nicely put into an open interval [math]x \in (a,b) \subset U[/math].

Br(x)={y:d(x,y)<r}B_{r}(x) = \{y : d(x,y) < r \}

(4,7)(4,7)

istheintervalofallrealnumbersstrictlybetween4and[math]7[/math]andisanopenset.Sois[math](4,2)(1,5)(7,)[/math]. is the interval of all real numbers strictly between 4 and [math]7[/math] and is an open set. So is [math](-4,2) \cup (1,5) \cup (7,\infty)[/math].

Sarnaselt komplektidele nagu

  1. Theemptysetandthefullset[math]X[/math]areincluded.Arbitraryunionsofsetsinthetopologyarealsointhetopology.Finiteintersectionsofsetsinthetopologyarealsointhetopology.The empty set \emptyset and the full set [math]X[/math] are included.Arbitrary unions of sets in the topology are also in the topology.Finite intersections of sets in the topology are also in the topology.

 on suletud.

Intervallid, kus üks külg on avatud ja teine ​​on suletud, ei ole avatud ega suletud.

Üldisemalt, kui meetrilises ruumis on avatud komplekt komplekt, kus kõigil selle elementidel on hingamisruum. Komplekti iga punkti jaoks võib selle keskel olla avatud pall, mis sisaldub komplektis. See on

UU

openxU  r:Br(x)U open \Leftrightarrow \forall x \in U \; \exists r : B_{r}(x) \subset U

Kui avatud pall on kõigi punktide kogum, mis on väiksem kui mõni vahemaa (raadius).

Br(x)={y:d(x,y)<r}B_{r}(x) = \{y : d(x,y) < r \}

Kui komplekt sisaldab mõnda oma piiri osa, ei saa see olla avatud. Kui komplekt ei sisalda ühtegi selle piiri osa, ei saa seda sulgeda.

Isegi üldisemalt on avatud komplektid kõik, mida me tahame, et nad oleksid nii pikad, kui nad järgivad topoloogia reegleid.

Topoloogia on kogum komplektidest, mida peame avatuks ja millel on järgmised omadused

  1. Tühi komplekt
  2. \emptyset
  3. andthefullsetXareincluded.Arbitraryunionsofsetsinthetopologyarealsointhetopology.Finiteintersectionsofsetsinthetopologyarealsointhetopology. and the full set X are included.Arbitrary unions of sets in the topology are also in the topology.Finite intersections of sets in the topology are also in the topology.

Meetrilised ruumid indutseerivad topoloogia, kasutades meie alusena avatud palle.


Vastus 2:

Suletud komplektil on miinimum ja maksimum; avatud komplekt seda ei tee.

Takeallofthenumbersbetweenzeroandone:[0,1].Thesmallestnumberinthesetiszero;thelargestnumberinthesetisone.Thesetisclosed.Take all of the numbers between zero and one: [0,1]. The smallest number in the set is zero; the largest number in the set is one. The set is closed.

Now,letsconsideraverysimilarset:[0,1).Theonlydifferencebetweenthissetandthelastoneisthatthisonedoesntinclude[math]1[/math].Itincludesnumbersthatarecloseto[math]1[/math];butitdoesntinclude[math]1[/math].Asaresult,itdoesnthaveamaximum:foranynumberthatsintheset,youcanfindanothernumberthatsalsointhesetbutislarger.Sincethereisnomaximum,thesetisopen.Now, let's consider a very similar set: [0,1). The only difference between this set and the last one is that this one doesn't include [math]1[/math]. It includes numbers that are close to [math]1[/math]; but it doesn't include [math]1[/math]. As a result, it doesn't have a maximum: for any number that's in the set, you can find another number that's also in the set but is larger. Since there is no maximum, the set is open.

Similarargumentsholdfor(0,1]and[math](0,1)[/math],whicharealsoopensets.Similar arguments hold for (0,1] and [math](0,1)[/math], which are also open sets.

Arelatedconceptiswhetherornotasetisbounded.Aboundedsethasasupremumandaninfimum:thesupremumofasetisthesmallestnumberthatsgreaterthanorequaltoeverythingintheset,andtheinfimumisthebiggestnumberthatslessthanorequaltoeverythingintheset.Unliketheminimumandmaximum,thesupremumandinfimumdonthavetobeintheset.InallfourcasesIhaveabove,1isthesupremumand[math]0[/math]istheinfimum.Allfourof[math][0,1][/math],[math][0,1)[/math],[math](0,1][/math],and[math](0,1)[/math]arebounded.Bycontrast,[math][0,)[/math]isanexampleofanunboundedset:thereisnonumberthatsgreaterthaneveryelementintheset,soithasnosupremum.Withoutasupremum,thesetcannotbebounded.A related concept is whether or not a set is bounded. A bounded set has a supremum and an infimum: the supremum of a set is the smallest number that's greater than or equal to everything in the set, and the infimum is the biggest number that's less than or equal to everything in the set. Unlike the minimum and maximum, the supremum and infimum don't have to be in the set. In all four cases I have above, 1 is the supremum and [math]0[/math] is the infimum. All four of [math][0,1][/math], [math][0,1)[/math], [math](0,1][/math], and [math](0,1)[/math] are bounded. By contrast, [math][0,∞)[/math] is an example of an unbounded set: there is no number that's greater than every element in the set, so it has no supremum. Without a supremum, the set cannot be bounded.

[0,1)[0,1)

.Theonlydifferencebetweenthissetandthelastoneisthatthisonedoesntinclude1.Itincludesnumbersthatarecloseto[math]1[/math];butitdoesntinclude[math]1[/math].Asaresult,itdoesnthaveamaximum:foranynumberthatsintheset,youcanfindanothernumberthatsalsointhesetbutislarger.Sincethereisnomaximum,thesetisopen.. The only difference between this set and the last one is that this one doesn't include 1. It includes numbers that are close to [math]1[/math]; but it doesn't include [math]1[/math]. As a result, it doesn't have a maximum: for any number that's in the set, you can find another number that's also in the set but is larger. Since there is no maximum, the set is open.

Sarnased argumendid kehtivad

(0,1](0,1]

and(0,1),whicharealsoopensets. and (0,1), which are also open sets.

Seotud mõiste on see, kas komplekt on piiratud või mitte. Piiratud komplektil on supremum ja infimum: komplekti supremum on väikseim arv, mis on suurem või võrdne kõige komplektiga ja infimum on suurim arv, mis on väiksem või võrdne komplekti kõigega. Erinevalt miinimumist ja maksimumist ei pea üla- ja alammäär komplektis olema. Kõigil neljal juhul, mis mul ülal on,

11

isthesupremumand0istheinfimum.Allfourof[math][0,1][/math],[math][0,1)[/math],[math](0,1][/math],and[math](0,1)[/math]arebounded.Bycontrast,[math][0,)[/math]isanexampleofanunboundedset:thereisnonumberthatsgreaterthaneveryelementintheset,soithasnosupremum.Withoutasupremum,thesetcannotbebounded. is the supremum and 0 is the infimum. All four of [math][0,1][/math], [math][0,1)[/math], [math](0,1][/math], and [math](0,1)[/math] are bounded. By contrast, [math][0,∞)[/math] is an example of an unbounded set: there is no number that's greater than every element in the set, so it has no supremum. Without a supremum, the set cannot be bounded.


Vastus 3:

Komplekti teooria, geomeetria või sellega seotud matemaatika harude korral aitavad avatud ja suletud komplektid kindlaks teha, kas mõni arv, mille kohta me uurime, on antud komplekti osa või mitte. Lühike seletus on ...

Ülaltoodud numbrireal tähistab O avatud komplekti, O = {x: x kuulub komplekti Z, x <5} (musta värviga joonistatud)

& C tähistab selle suletud komplekti, C = {x: x kuulub Z, x>, = 5} (sinise joonisega)

Siin on vaja seda mõista. Kui räägime mis tahes komplektist, on see täpselt määratletud objektide kogum, st objektidel on mõned sarnased omadused. Nagu E, paarisarvude komplekt. Z, täisarvude komplekt. jne… MITTE mõnikord tekitab probleem neis piire. Ja see on punkt, kus vajame avatud ja suletud komplekti.

Avatud komplektis, x <5, siin me piirpunkti 5. ei lisa. Niisiis, väike ring tõmmatakse ümber 5. Teil on lubatud minna nii lähedale, kui soovite oma limiidile 5. Kuid te ei saa kunagi puudutada 5

Arvestades, et selle suletud komplekt, x> = 5, mis sisaldab kõiki neid elemente, mida O-s pole, ja lisaks on eeliseks see, et see võib puudutada 5. st C sisaldab limiiti 5. st määratud keeles on suletud komplekt täiend avatud komplekt.

Veel üks näide:

Ülaltoodud numbrireal avatud komplekti jaoks {x <2; x> 5}, suletud komplekt on

{2 <, = x <, = 5}

Ülaltoodud numbrireal on avatud komplekti {2

{x <, = 2; x>, = 5}