-Space, subspace관련
-어떤 subsets을 포함하는 가장 작은 top생각가능(즉, top의 intersection은 top됨)
-top들의 collection에 의해 generated top도 생각가능
-Order Topology관련
-T4(따라서 T3,T2 이런 것도 다 됨)
-least upperbound property가 성립 iff 모든 closed interval(not singleton)은 compact
-linear continuum가 성립 iff TS가 connected
-linear continuum이 성립할 때 구체적으로는
-V는 connected이고 따라서 전체집합, intervals, rays모두 connected됨
-locally compact
-T4
-well ordered인 경우(least upper bound가 성립)
-T5
-Subspace관련
-From TS to S
-모든 S에 대하여
-strict total order relation
-open
-closed
-basis
-closure(E<S, cl(E) in S =cl(E) in TS intersection S)
-T2
-T2.5
-T3
-T3.5
-CN
-T5
-f:X->Y, conti, S<X이면 g:S->Y도 conti
-f:X->Y, conti, f(X)<S1<Y이면 g:X->S1도 conti
-first-countability
-second-countability
-covering map(f:TS1->TS2, covering map일 때 TS2의 subspace S2를 잡고,f:f^(-1)(S2)->S2가 covering map도 된다는 것)
-S가 open in TS일 때만
-LKT2(LK만 되는지는 모름)
-S가 closed in TS일 때만
-compact
-paracompact
-LK
-lindelof
-T4
-기타
-S with induced order은 S as subspace랑 다르다. (S가 convex in TS이면 가능)
-From S to TS
-모든 S에 대하여
-f:X->S, conti, S<Y이면 g:X->Y도 conti
-S가 open in TS일 때만
-open
-S가 closed in TS일 때만
-closed
-NTS
-Product관련
-Prod(S_i) = subspace of Prod(TS_i)
-From TS_i to Prod(TS_i) (곱이 countable개이냐 아니냐/product top이냐 box top이냐 구분)
-open(box top에서만 됨)
-closed
-basis
-T2
-T3
-T3.5(product top에서만 됨)(link)
-f:TS->Prod(TS_i)의 conti(product top에서만 됨)
-seq의 수렴성(product top에서만 됨)
-connected(product top에서만 됨)
-(Tychonoff's Theorem)compact(product top에서만 됨, product를 uncountable개 하더라도)
-path-connected(product top에서만 됨)
-TS1,TS2:path-connected일 때 FHG(TS1xTS2,x1,x2) giso FHG(TS1,x1)xFHG(TS2,x2)
-Countable Prod일 때
-first-countability
-second-countability
-separable
-metrizable
-complete
-totally bdd(각 TS_n가 MetricS)
-Finite Prod일 때
-covering map
-From Prod(TS_i) to TS_i
-Using projection
-open
-f:TS->Prod(TS_i)의 conti
-seq의 수렴성
-metrizable
-connected
-T2
-T3
-T4
-Quotient Space, QS(TS,~)관련
-From TS to QS(TS,~)
-TS:connected이면 QS(TS,~):connected
-TS:path-connected이면 QS(TS,~):path-connected
-TS:locally connected이면 QS(TS,~):locally connected(link)
-TS:compact이면 QS(TS,~):compact
-TS:T3이고 E:closed in TS이면 QS(TS,E):T2(T3가 아니라 T2가 맞음)
-TS:T4이고 E:closed in TS이면 QS(TS,E):T4
-From QS(TS,~) to TS
-QS(TS,~):connected, 각 class가 connected in TS이면 TS도 connected
-LKT2관련
-LKT2의 해석방법
-compact nbd
-pre-K open set
-TS1:LKT2 iff te TS2 s.t. TS1<TS2 and TS2-TS1 is singleton and TS2:KT2(link)
(TS1:LKT2일 때, TS2를 ocl(TS1)이라 표기하기로 하자. 왜냐하면 추가한 point가 TS1의 limit pt가 됨)
(TS2는 유일 up to homeomorphic)
(TS1:open subspace of ocl(TS1))
-E1:open containing x일 때, te E2:open containing x s.t. E2:pre-K, cl(E2)<E1(link)
-E1:open containing K일 때, te E2:open containing K s.t. E2:pre-K, cl(E2)<E1
-Baire
-CGT
-T3
-T3.5
-KT2관련(KT2도 일단 LKT2이므로 LKT2의 성질들 만족)
-isolated pt가 하나도 없으면 uncountable space이다.(link)
(isolated pt란, {pt}:open일 때, 그 pt를 isolated pt라 한다.)
-BaireS(link)
-LKT2됨(LKT2 성질들 다 만족)
-T3
-T4
-Metrizable iff second-countable
-CGT관련
-CGT의 예:LK, K, first-countable, MetricS
-f:CGT->TS가 for any K, restriction of f on K is conti이면 f는 conti
-Retract of TS관련
-TS:T2이면 Retract of TS:closed in TS(link)
-Classification of Surfaces
-Polygonal region을 pasting edges해서 얻은 space는 T2 compact connected surface가 된다.
(quotient는 compact,connected을 preserve하고, 원래 polygonal region이 2-dim이므로)
(finite polygonal regions이면 connected빼곤 다됨, quotient가 closed map이므로 T2 preserve함)
-만약 모든 vertex가 1개의 vertex로 mapping되면(by quotient), 얻어진 surface의 FHG는
FP(Z, label개수만큼)/<scheme을 한바퀴 돈것>
-two schemes가 equivalent란, 얻어진 quotients가 homeo일 때
-scheme연산(equivalent를 얻는) 것으로는
-cut
-paste
-relabel
-permute
-flip
-cancel
-uncancel
-한개의 polygonal region으로 만든 T2 compact connected surface는 homeo S2 or n-fold torus or m-fold projective plane
-compact surface관련
-모든 compact surface는 triangulable
-모든 connected compact surface는 하나의 polygonal region을 pasting edges해서 얻을 수 있다.
-따라서 connected compact surface는 classification됨
-{n-fold torus, n=1,2,3,..} bij {compact orientable Riemann Surfaces}
-Subset관련(open, closed, compact, connected, convex, dense...)
-operation on subset관련
-cl은 monotone, commute with finite union, product
-cl취해도 변하지않는 성질들
-connected
-diam
-E의 interior, exterior, boundary는 TS를 partition함, 이때 interior, boundary는 cl(E)를 partition함
-(cl(E))^c = int(E^c) / (int(E))^c = cl(E^c) (드모르간 법칙 같네)
(따라서 Baire Space의 정의를 closed sets이용해서 state할 수 있고, 혹은 open sets을 이용할 수도 있다.)
(E:dense in TS iff E^c has empty interior)
-Bd(E) = empty iff E:open and closed
-Limit point관련
-E1<E2, x:limit point of E1이면 x:limit point of E2
-Connected관련
-C가 connected TS란,
-there is no separation이 정의
(separation (U,V)란, U와 V가 disjoint, nonempty, open, union=TS을 가리킴)
-empty set is connected
-every singleton subset set is connected
-separation G1,G2에 대해 G1의 limit pt는 G2에 속하지 않는다. (G2의 limit pt는 G1에 속하지 않는다.)
(즉 Separation은 separated sets이다.)
-TS:connected iff te no separation
-TS:connected iff clopen sets은 TS랑 empty뿐
-G1,G2:separation of TS, S:connected이면 S<G1 or S<G2
-S_i가 connected일 때, union S_i도 connected(단 common pt가 있을 때)
-(Intermediate Value Theorem)f:C->TS with order top, conti이고 f(a)<r<f(b)이면 te c in C s.t. f(c)=r
-connected component는 closed이다.
(open일 때도 있는게, component가 유한개거나, locally connected이면 된다.)
-path-connected관련(connected임을 보이는 쉬운 방법중 하나)
-TS:path-connected이면 connected이다.(역은 성립 안함)
-S_i가 path-connected일 때, union S_i도 path-connected(단 common pt가 있을 때)
-path-connected components는 connected components에 포함된다.
(S:path-connected라해서 cl(S)가 path-connected인 것은 아니다.)
(path-connected component는 closed일 필요도 없고 open 일 필요도 없다.단, locally path-connected이면 open은 된다.)
-totally disconnected관련
-Compact관련
-empty set은 compact
-compact set is closed if TS:T2
-모든 singleton set은 compact
-TS:compact iff every collection of closed sets in TS having finite intersection property, intersection of all elements in the collection is nonempty
-finite union of compact is compact
-LK
-CGT
-(Tube Lemma)
-TS1xTS2, TS2:compact, N:open in TS1xTS2 containing x_0 x TS2이면 te E s.t. N contains ExTS2, x_0 is in E, E:open in TS1
(꼭 N이 x_0 x TS2를 contain할 상황 말고도 subset x TS2형태를 contain하더라도 적용가능, 얻은 E를 union해버리면 되므로)
-S1<TS1, S2<TS2, S1과S2 둘다 compact, N:open in TS1xTS2 containing S1xS2이면 te E1,E2 s.t. E1:open in TS1, E2:open in TS2, S1xS2 < E1xE2 < N(link)
(두번째 것이 첫번째 것을 포함하지만, 첫번째 것만으로도 자주 나오니 구분해서 적음)
-(Extreme Value Theorem):f:K->TS with order top, conti이면 te c,d in K s.t. f(c)<=f(x)<=f(d) for all x in K
-비슷한 compact관련
-compact이면 limit point compact이다.
-compact이면 lindelof이다.
-Metrizable일 땐, compact=limit point compact=sequentially compact
-limit point compact의 closed subset은 limit point compact
-paracompact관련
-TS:metrizable이면 paracompact
-TS:RTS and lindelof이면 paracompact
-TS:T2 and paracompact이면
-TS:T4
-for any finite open cover {E_i}, te a partition of unity on TS dominated by {E_i}
-smooth mnf는 paracompact
-Convex관련(오직 Strict Total Order Relation을 가진 E에서만 생각)
-Local Property관련
-Locally Connected관련
-TS:locally connected iff every connected components of every open in TS is open
-Locally Path-Connected관련
-TS:locally path-connected iff every path-connected components of every open in TS is open
-TS:locally path-connected이면 path-connected component=connected component, 게다가 open(link)
-Locally Compact관련
-E:compact->E:locally compact
-Locally Homeomorphic관련
-every homeomorphism is local homeomorphism
-local homeomorphism은 open map, conti(link)
-bijective local homeomorphism은 homeomorphism
-f:TS1->TS2, local homeomorphism일 때 preserve하는 성질들
-TS1이 locally path-connected이면 f(TS1)도 locally path-connected
-TS1이 locally connected이면 f(TS1)도 locally connected
-TS1이 locally compact이면 f(TS1)도 locally compact
-TS1이 first-countable이면 f(TS1)도 first-countable
-Countability관련
-first-countable관련
-TS:가 first-countble이고 E<TS일 때 x:limit point of E이면 te {x_n} in E s.t. cv to x(역은 first-countability아니어도 성립)
-f:X->Y, X가 First-Countability이면 Sequentially conti->conti
-TS:first-countable이면 TS는 CGT
-Second-Countable관련
-TS:second-countable이면 모든 discrete subspace는 countable이다.
-second-countable이면 separable이다.(metrizable이면 역도 성립)
-second-countable이면 lindelof이다.(metrizable이면 역도 성립)
-second-countable이면 first-countable이다.
-TS:second-countable이고 E:uncountable이면 uncountable many pts in E는 E의 limit pt이다.
(즉 second-countable이면 uncountable E들은 limit pts를 uncountable개 갖는다 in E)
-Lindelof
-countable TS이면 lindelof이다.
-compact이면 lindelof이다.
-Separable
-TS:separable이면 every collection of disjoint open sets는 countable
-Separation관련
-포함관계
-T0>T1>T2>T2.5>CT2>T3>T3.5>T4>T5>T6
-R>CR
-N>CN>PN
-T0(2pt, topologically distinguishable)관련
-T1(2pt, separated, separated란 each가 cl(the other)와 disjoint, or open sets으로도 해석 가능)관련
-(iff)모든 finite set은 closed
-E의 limit L iff open(L) intersection E는 infinitely many pts을 포함
(first-countability는 L을 포함하는 open set의 개수가 countable개 있음을 보장해주고, T1은 intersection의 원소가 무한개임을 보장해준다.)
-T2(2pt, separated by open nbd)관련
-seq의 limit은 기껏해야 1개
-TS:T2S iff TSxTS의 diagonal은 closed in TSxTS
-compact subspace는 closed됨
-compact와 pt는 separated by open nbd
-2 compact는 separated by open nbd
-f:TS1->TS2 conti, g:TS1->TS2 conti, TS2:T2일 때, {x in TS1|f(x)=g(x)}는 closed in TS1
(따라서 f:TS->TS conti, TS:T2일 때, fixed points의 모임은 closed in TS됨)
-T2.5(2pt, separated by closed nbd)관련
-CT2(2pt, separated by conti function)관련
-R(closed와 pt, separated by open nbd)관련
-(iff)closed와 pt에 대해 separated by closed nbd
-(iff)for any x in TS, any open U containing x, te open V containing x s.t. cl(V)<U
-(T1도 되면)T3라 한다.(T0,T1,T2 중 어느것이 되도 상관없음, 이후 일관성 때문)
-TS:T3이고 lindelof이면 T4
-TS:T3이고 second-countable이면 T4, CN, T5, metrizable, imbedded in R^N(product top or uniform top)
(metrizable임을 보일 때, TS가 imbedded in R^N (with product top or uniform top)임을 보인다.)
-CR(closed와 pt, separated by conti function)관련
-closed와 compact가 주어지면 separated by conti function 가능
(단, closed와 closed일 때까지로는 확장 못함)
-(T1도 되면)T3.5라 한다.(T0,T1,T2 중 어느 것이 되도 상관없음)
-TS:T3.5 iff TS homeo S of [0,1]^J for some J.
([0,1]^J 는 KT2됨)
-TS:T3.5 iff TS has a compactification
-TS:T3.5이면 te! SCcl(TS) up to homeo s.t. for any f:TS->KT2, conti, f can be uniquely extended to SCcl(TS), conti.
-N(2closed, separated by open nbd)관련
-TS:NTS
iff for any closed set E1 in TS, any open E2 s.t. E1<E2, te open E3 containing E1 s.t. cl(E3)<E2(link)
iff (Urysohn's Lemma)2closed, separated by conti function(link1)(link2)
iff (Tietze Extension Theorem)for any closed E in TS, for any conti f:E->[0,1],
f has extension g:TS->[0,1], conti, restriction of g on E=f(link)
(Tietze Extension Theorem에서 [0,1]대신 [a,b], (a,b), R(std), [0,1]^n형태도 가능)(link)
-(T1도 되면)T4라 한다.(T0는 안됨, T1,T2 중 어느 것이 되도 상관없음)
-TS:T4이고 connected이면 |TS|=1 or uncountable이다.(확인필요 꼭 T4여야하는지)
-T4
-(Existence of Partitions of Unity)for any finite open cover {E_i}, te a partition of unity on TS dominated by {E_i}
-CN(2separated sets, separated by open nbd)관련
-(iff)모든 subspace가 N
-(T1도 되면)T5라 한다.(T0는 안됨, T1,T2 중 어느 것이 되도 상관없음)
-PN(2closed, precisely separated by conti function)관련
-(iff)모든 closed set이 Gd
-(T1도 되면)T6라 한다.(T0는 안됨, T1,T2 중 어느 것이 되도 상관없음)
-Sequence관련
-Directed Set, Net, Net Convergence관련
-Net convergence를 도입시 좋은 점
-x in cl(E) iff te seq {x_n} in E cv to x(단 TS가 first-countable일 때 only if가 성립-(*))
-f:sequentially conti iff f:conti(단 domain이 first-countable일 때 only if가 성립-(*))
-TS:sequentially compact iff TS:compact(단 TS가 Metric일 때 only if가 성립-(*))
((*)에서 seq를 net으로 바꾸면 단~~ 부분이 필요없어지게 된다.)
-Map관련(Conti, Homeo, Open map, closed map, quotient map, Projection,)
-Projection의 성질
-open map(closed map는 아닐 수 있음)
-conti
-not closed(TS1xTS2->TS1에서 TS2가 compact라면 closed map됨, (link))
-conti criteria, f:X->Y
-X의 open개수가 많고 Y의 open개수가 적을수록 conti될 가능성이 높아짐
-Use closed in Y
-Use open in Y
-Use basis in Y
-Use subbasis in Y
-f(cl(E))<cl(f(E))
-Using Pasting lemma, open sets in X(uncountable개여도 상관없음)
-Using pasting lemma, finite closed sets in X
-f가 conti이면 sequentially conti(역은 domain TS가 first-countability필요)
-(Closed Graph Theorem in TS)Y:KT2일 때는, f가 conti iff the graph of f is closed in XxY가 성립(if는 K인걸 이용, only if는 T2인 걸 이용)(link)
-open map, closed map, continuous map, quotient map의 성질(f(TS1)에서의 성질들이다 continuous image관련해서는)
-f:TS1->TS2, conti,
-f:(TS1,C4(TS1))->(TS2,C4(TS2))가 conti면 MF도 된다.
-f:S(<TS1)->TS2, TS2:T2일때, extension of f on cl(S)는 unique(link)
-f(connected)=connected
-f(compact)=compact
-f(path-connected)=path-connected
-f(lindelof)=lindelof
-f(dense)=dense
-f(separable)=separable
-f:open이기도 하면
-f(basis)=basis of f(TS1)
-f(locally compact)=locally compact
-f(first-countable)=first-countable
-f(second-countable)=second-countable
-f:surjective이면 f:quotient map
-f:closed이기도 하면
-f(T4):T4(link)
-f:surjective이면 f:quotient map
-TS2:order top이고, g:TS1->TS2, conti일 때, {x in TS1|f(x)<=g(x)}:closed in TS1, h=min(f,g):conti(link)
-f:quotient map일때
-restriction of quotient map to class or union of classes
-class or union of classes가 open혹은 closed였으면 restriction도 quotient
-quotient map이 open or closed map이었으면 restriction도 quotient
-f:injective이기도하면 f:homeomorphism
-(Characteristic Property of Quotient Map)
:g:TS2->TS3:conti iff g o f:conti
-f:TS1->TS2, closed
-TS1:T1이면 f(TS1):T1
-U:open in TS1, E2:subset in TS2, s.t. f^(-1)(E2)<U이면 te V:open in TS2 s.t. E2<V, f^(-1)(V)<U(link)
-proper map관련
-TS2가 first-countable이고 pt마다 pre-K nbd를 가지고 f:proper conti이면 f는 closed
-f:TS1->TS2, proper되기위한 충분조건
-TS1:compact and TS2:T2이고 f가 conti이면 f는 proper
-f가 proper이고 E<TS1이 saturated wrt f일 때 restriction of f on E는 proper
-TS1,TS2가 T2이고 f가 conti이고 te g:left inverse of f s.t. g:conti이면 f는 proper
-Topological Properties관련
-모음
-connectedness
-compactness
-local connectedness
-metrizability
-first-countable
-second-countable
-lindelof
-separable
-fundamental group(giso일 듯? 이후 수정)
-Compactification관련
-TS has a compactification TS2이면 TS는 T3.5
-Functions Collection관련(혹은 Functions Seq관련)(Domain과 Range에 Metric이 없어도 되는 경우)
-fC(J,TS)에서(top of pt cv정의가능)
-seq {f_n} in fC(J,TS) cv in the top of pt cv iff {f_n}:pt cv.(fC(J,TS)의 product top과도 같음)
-fC(TS1,TS2)에서
-fCconti(TS1,TS2)에서(KG-top정의가능)
-top of pt cv < KG-top
-eval:LKT2 x fCconti(LKT2,TS) -> TS는 conti
-With Measure
*Algebraic Topology+Differential Geometry
-Homotopy, Homotopy of Paths관련
-about =_homotopic
-equivalence relation을 만든다
-f1 =_homotopic f2 of TS1 into TS2, g1 =_homotopic g2 of TS2 into TS3일 때 g1(f1) =_homotopic g2(f2)(link)
-TS1->R^2(std)인 경우
-straight-line homotopy를 통해서 임의의 conti function f1,f2가 =_homotopic임을 알 수 있다.
-특히 f1, f2의 image가 convex set에서만 생긴다면 straight-line homotopy의 image모두 convex set안에 유지
-about [TS1,TS2]
-[TS1, [0,1]] has a single element(link)
-[[0,1], path-connected TS] has a single element(link)
-TS2:contractible이면 [TS1,TS2] has a single element(link)
-TS2:path-connected더라도(contractible보단 약한), TS1:contractible이면 [TS1,TS2] has a single element(link)
-about contractible
-contractible TS는 path-connected이다.(link)
-TS1:contractible iff for any TS2, for any f:TS2->TS1, g:TS2->TS1, 둘다 conti, f =_homotopic g
-TS:contractible이라해서 strong deformation retract singleton이 존재하는건 아님, zigzag예 생각
-about =_phomotopic(1개의 TS의 성질 관심)
-equivalence relation을 만든다.
-path1의 final이 path2의 initial인 paths끼리 product연산(*)을 줄 수 있다.
(별말없이 path1*path2라 썻다면 path1의 final=path2의 initial인 상황)
-[path1]*[path2]=[path1*path2]로 정의하면 phomotopic classes에 product연산(*)을 줄 수 있다.
-phomotopy path1,path2(in TS1)에 conti인 k:TS1->TS2를 합성하면, phomotopy k(path1), k(path2)가 된다.
-conti인 k:TS1->TS2가 있을 때 k(path1*path2)=k(path1)*k(path2)가 성립
-product연산 on phomotopic classes은 groupoid properties를 만족(link)
(group axioms가 성립하지 않는 유일한 다른점은 final과 initial이 같은 path classes사이에서만 연산된다는 점)
-path in TS를 n개의 path로 쪼개기 가능, [path]=[path1]*[path2]*...*[pathn]
-[0,1]->R^2(std)인 경우
-straight-line homotopy를 통해서 임의의 path1, path2 with same initial, final가 =_homotopic임을 알 수 있다.
-특히 path1, path2의 image가 convex set에서만 생긴다면 straight-line homotopy의 image모두 convex set안에 유지
-R^2(std)-{0}에서는 성립안함. UO1에서 시계방향, 반시계방향 path가 not phomotopic
-About Fundamental group(FHG(TS,x))
-FHG(TS,x)는 group이 된다.
-path from x to y가 있다면 FHG(TS,x)->FHG(TS,y)인 giso를 얻을 수 있다.
-TS가 path-connected이면 for any x, y in TS FHG(TS,x) giso FHG(TS,y)
(TS가 path-connected인 경우에 Fundamental Group에 대해서 이론전개하는 이유이다.)
(TS가 path-connected이면 FHG(TS,x)에서 base point 언급이 필요없을 것 같지만, base point 사이 path결정에 따라 giso가 달라지므로 일반적으로 TS가 path-connected여도 base point언급을 한다.)
-E:path-connected component(subspace)인 경우 FHG(E,x)=FHG(TS,x)가 된다.
-TS:path-connected이면 for x in TS, FHG(TS,x)가 abelian iff giso from FHG(TS,x) and FHG(TS,y)는 unique(link)
-TS:simply connected인 경우
-FHG(TS,x)=trivial
-TS의 임의의 두 paths with same initial, final인 path1, path2는 phomotopic이다.
-FHG은 topological property이다.
(homeo:TS1->TS2, homeo(x)=y일 때 FHG(TS1,x) giso FHG(TS2,y))
-(Pre Van-Kampen Theorem)(link)
:if TS=union of E_i, E_i:path-connected and open, E_i교E_j:path-connected, te x in all E_i, then any loop in X based at x is homotopic to a product of loops based at x each of which is contained in E_i
-(Van-Kampen Theorem)
:if TS=union of E_i, E_i:path-connected and open, E_i교E_j:path-connected, te x in all E_i,
then the homog F:FP(FHG(E_i,x))->FHG(X,x) is surjective where i_i:E_i->X, inclusion이므로 homog from i_i인 f_i:FHG(E_i,x)->FHG(X,x)이고 Universal mapping property of free product에 의해 만든 F
그리고 E_i교E_j교E_k for any i,j,k가 path-connected이면 ker(F)=the normal subgroup of FP(FHG(E_i,x)) generated by all elements of the form i_(ij)(w)i_(ji)(w)^(-1) for w in FHG(E_i교E_j,x), where i_(ij)와 i_(ji)는 inclusions:E_i교E_j->E_i, :E_j교E_i->E_j에 의해 induced homog
-(Adjoining 2-cells)(link)
-따라서 어떤 TS의 FHG를 구하고 싶을 때
-covering map
-deformation
-van-kampen
-scheme을 구한 다음, label개수만큼 UO1을 wedge product한 다음, scheme을 adjoin해서 구할 수 있다.
-FHG(P(R^2))=Z/2Z
-FHG(2-torus)=ZxZ
-FHG(Klein bottle)=FP(Z,Z)/<aabb> where Klein bottle:aba^(-1)b를 scheme으로 가지는 것
-(FHG Functor is surjective)For any G:group, te X:TS s.t. FHG(X)=G using presentation of G and adjoing 2-cells
-About homo from (h,x_0)
-h1:TS1->TS2, h1(x_0)=y_0, h1:conti이고 h2:TS2->TS3, h2(y_0)=z_0, h2:conti일 때,
homo from (h2 o h1, x_0)=homo from (h2,y_0) o homo from (h1,x_0)
-homo from (identity,x_0)는 identity group homomorphism이다.
-h가 homeomorphism일 땐 homo from (h,x_0)는 giso가 된다.
-h:TS1->TS2, h:conti이고 TS1:path-connected였다면 homo from (h,x_0)에서 x_0로 뭘 택하든 같은 homomorphism을 얻는다.
(path-connected까진 아니어도 TS1에서 path가 있었다면... 다음 참고 link)
-About Same homotopy type
-Homeomorphic한 2 spaces끼린 same homotopy type
(역은 성립안함, same homotopy type을 가진다 하더라도 homeo하진 않을 수 있다.)
-{TS_i}에서 same homotopy type이란 relation을 주면 equivalence relation된다.
-Fiber bundle관련
-(E,p,B):F-bundle일 때
-p^(-1)(b) homeo F
-p:open
-따라서 p는 quotient map(근데 이게 원래 정의보다 강력하지 않음, 그냥 open conti surjective로 아는게 도움됨)
-for any open V in B, (F, p^(-1)(V), restriction of p on p^(-1)(V), V)또한 f-bundle된다.
-(E,p,B):R^k(std)-vector bundle일 때
-(conti or smooth)local frame for E over V가 있으면 local section on V가 (conti or smooth)한지 판단가능, using component functions wrt given local frame(link)
-vf_U의 경우는 local frame for TM over U를 coordinate chart on M과 R^n(std)상의 partial derivatives의 inverse로 이용해서 판단가능
-cvf_U의 경우는 위의 local frame의 dual basis로 이용해서 판단가능
-(M1,p,M2):smooth R^k(std)-vector bundle일 때
-M:smooth manifold일 때, smooth R^k(std)-vector bundle만드는 방법(link)
-transition function의 정의와 성질, link참고(link)
-(M1,p1,M), (M2,p2,M)사이의 smooth bundle map f:M1->M2는 다음을 만족한다.
-f:SGS1(M)->SGS2(M)은 C^inf(M)-linear
-g:SGS1(M)->SGS2(M)이 C^inf(M)-linear이면 te smooth bundle map f:M1->M2 s.t. f=g
-p:submersion
-SGS(M2):VS(R(std)) and C^inf(M2)-Md
-(Extension of smooth local section over closed to global)for A:closed in M2, f:A->M1 s.t. smooth and section of p, U:open containing A일 때 te g in SGS(M2) s.t. g=f on A and support of g < U
-{Smooth local frame} bijective {Smooth local trivialization}(->가 어려움)(link1)(link2)
-(M1,p,M2):trivial iff it admits a smooth global frame
-for a (smooth)coordinate chart (V,g) for M2, smooth local frame for M1 over V, te a (smooth)coordinate chart (p^(-1)(V),f) for p^(-1)(V)(link)
(즉, (M1,p,M2)에서 M2에서 좌표잡겠다고 (smooth) coordinate chart on V갖고오면 p^(-1)(V)에서도 좌표논의 가능)
-(TM,p,M)관련
-vf:smooth iff for any U:open in M, for any f in C^inf(U), vf_U(f):smooth
-f:C^inf(M)->C^inf(M):derivation iff f is of the form vf_M for some smooth vector field vf
-srv-bundle of rank n where n=dim(M)
-transition function은 Jacobian matrix가 된다.
-M:parallelizable iff (TM,p,M):trivial
-VF(M), derivation의 collection으로 보면
-for F:M1->M2, smooth, vf1 in VF(M1), vf2 in VF(M2)일 때, vf1,vf2:F-related iff vf1(f o F)=vf2(f) o F for any f in C^inf(V), V:any open in M2
(즉 F-related판정을, vf를 derivation으로 보아 판정 가능)
-for F:M1->M2, diffeo, vf1 in VF(M1), te! vf2 in VF(M2) s.t. vf1,vf2:F-related(vf2(F(p))=pf_p(F)(vf1(p)))
-Lie R-A된다.
-for vf1,vf2 in VF(M), [vf1,vf2]의 계산은, 각각을 먼저 coordinate로 표현하면 쉬워진다.(link)
-Lie R-A이면서 C^inf(M)-Md인 걸 생각하면 다음 공식얻는다.
for f,g in C^inf(M), vf1,vf2 in VF(M), [fvf1,gvf2]=fg[vf1,vf2]+(fvf1g)vf2-(gvf2f)vf1(link)
-for F:M1->M2, smooth, vf1,vf3 in VF(M1), vf2,vf4 in VF(M2), vf1,vf2:F-related and vf3,vf4:F-related일 때 [vf1,vf3],[vf2,vf4]:F-related
-(CTM,p,M)관련
-cvf:smooth iff for any U:open in M, any vf_U, (cvf,vf_U):U->R(std)가 smooth
-local frame for M over U가 있으면 local coframe을 만들 수 있다. using duality 이 때 얻은 coframe을 dual coframe to the given frame이라 함.
-for U:open in M, x in M, f in C^inf(U), df_x는 smooth cvf_(U,x)이다.
-(Properties of df)(U:open in M, f,g in C^inf(U), a,b in R(std),
-d(af+bg)=adf+bdg
-d(fg)=fdg+gdf
-d(f/g)=(gdf-fdg)/g^2 on the set where g != 0
-if Im(f)<J, J:interval in R(std), h in C^inf(J), then d(h o f) = (h' o f) df
-df=0 iff f is constant on each component of M
-df_p(v) is the best approximation f(p+v)-f(p)
-f in C^inf(M), r:J->M, smooth curve in M일 때 (f o r)'(t) = df_(r(t))(r'(t))(link)
-for F:M1->M2, smooth, cvf2 in CVF(M2)일 때 we can define cvf1 in CVF(M1) s.t. cvf_p = pb_(F(p),F)(cvf2_F(p))
(cvf2가 smooth이면 cvf1도 smooth하게 가능)
-for F:M1->M2, smooth, g in C^inf(M2), h in CVF(M2)(link)
-pb_(F)(dg)=d(g o F)
-pb_(F)(gh)=(g o F) pb_(G)(h)
-cvf:conservative iff cvf:exact
-cvf:exact이면 cvf:closed(따라서 exacteness체크 쉬움, 게다가 모든 charts가 아니라 M을 cover하는 charts collection에 대해서만 보여도 충분함)
-cvf:closed이면 exact이다는 domain(M)의 모양에 depends
-star-convex open in R^n(std)이면 성립
-every closed cvf is exact on any simply connected manifold
-(Local Exactness of Closed Covector Fields)cvf:closed on M이면 every p in M has a nbd(p) on which cvf:exact
-for f:M1->M2, local diffeo일 때 pb_(f)은 closed cvf를 closed cvf로, exact cvf를 exact cvf로 mapping
-cvf:exact일 때 potential은 여러개 있을 수 있으나 상수만큼만 차이남
-Covering관련
-TS1:covering space of TS2 with covering map f:TS1->TS2일 때, for any y in TS2, f^(-1)(y) has a discrete topology induced from TS1
-covering map f:TS1->TS2이면
-surjective
-continuous
-f:local homeomorphism
-f:open map
-g:TS2->TS3가 covering map이고 g^(-1)(z):finite set for all z in TS3이면 f o g도 covering map(link)
-f(x1)=x2이고
-(Existence of Lift(path), uniqueness)path:[0,1]->TS2, path(0)=x2일 때 te! lift(path) to TS1 s.t. lift(path):a path and lift(path)(0)=x1(link1)(link2)
-(Existence of Lift(phomotopy), uniqueness)F:[0,1]x[0,1]->TS2, conti, F(0,0)=x2일 때 te! lift(F) to TS1 s.t. lift(F):conti and lift(F)(0,0)=x1
(F가 phomotopy였으면 lift(F)도 phomotopy됨)
-path1:[0,1]->TS2, path2:[0,1]->TS2가 phomotopic이면 lift(path1) to TS1, lift(path2) to TS1도 photomopic
-TS1:path-connected이면 lifting correspondence_(f,x1):FHG(TS2,x2)->f^(-1)(x2)은 surjective이다.(link)
-TS1:simply connected이면 liffting correspondence_(f,x1):FHG(TS2,x2)->f^(-1)(x2)은 bijective이다.(link)
-homo from (f,x1)은 injective이다.
-S:=homo from (f,x1)(FHG(TS1,x1)), lifting correspondence_(f,x1) induce g:FHG(TS2,x2)/S->f^(-1)(x2) and g:injective
(TS1:path-connected이면 g:bijective까지도 됨)
-l:loop in TS2 based x2일 때 [l] in S iff lift(l) to TS1은 a loop in TS1 based x1
-가장 쉬운 예: f:R(std)->UO1(subspace of R^2(std)), f(t)=(cos(2*pi*t),sin(2*pi*t))
-From TS1 to TS2
-
-From TS2 to TS1
-T2
-T3
-T3.5
-locally compact hausdorff
-compact(단, covering map f:TS1->TS2, f^(-1)(y):finite for all y in TS2인 경우만)
-Universal covering관련
-(Universal Covering이라 부르는 이유)TS1:universal covering space of TS2일 때, for any g:TS3->TS2 covering map, te h:TS1->TS3 covering map
(즉 Universal covering space는 다른 covering space를 cover한다. 즉 cover 중에서 가장 넓은 개념)
-(Existence of Universal Covering Space)if TS:connected and locally simply connected이면 te universal covering space of TS
-Retraction관련
-retraction(TS,S):conti일 때
-quotient map이 된다.
-S를 retract of TS라 부른다.
-inclusion:S->TS를 f라 하면 homo from (f,x):injective for any x in S
-Topological Group관련(*는 group의 operation으로보자)
-자주 쓰이는 함수관련
-f:TGxTG->TG, f(g1,g2)=g1*(g2)^(-1):conti
-conjugation by g, left multiplication by g, inversion, 3개다 TG->TG인 homeo
-{all SME(e)}는 nbd basis됨, 즉 for any nbd(e), te nbd2(e) in {SME s.t. SME=nbd(e)} s.t. nbd2(e)<nbd(e)
-left multiplication by any element가 homeomorphism이므로 {all SME(e)}만 이용하면 많은 문제 해결됨
-TG가 discrete top을 가진다 iff {e}:open
(포인트는 이 명제 자체가 아니라, TG에서는 nbd(e)에 대해서만 생각하면 된다는 것이 포인트)
-for any g in TG, for any nbd(g), te SME(e) s.t. SME(e)*g*SME(e) < nbd(g)
-for any nbd(e) and n:자연수, te SME(e) s.t. (SME(e))^n < nbd(e)
-For any subset E, cl(E)=intersection E*SME(e) over all SME(e)=intersection SME(e)*E over all SME(e)=intersection SME(e)1*E*SME(e)2 over SME(e)1, SME(e)2
-cl(subTG):subTG(subgroup이 된다는 게 포인트)
-cl(NS):NS(normal이 유지된다는 게 포인트)
-Topological Manifold관련
-smoothness는 not topological property
-T2가 정의상 필요한 이유는 TS에서 T2가 필요한 것과 마찬가지, limit의 유일성, finite set의 closedness가 필요
-Second-countable이 필요한 이유는 partitions of unity의 존재성보일 때 등등
-top n-mnf이 동시에 top m-mnf이 될 수는 없다.(Invariance of Domain참고)
-모든 top n-mnf
-has a countable basis of pre-K coordinate balls
-locally compact
-locally path-connected(이것이므로 아래 3개가 성립)
-locally connected
-top n-mnf:connected iff top n-mnf:path-connected
-path-connected components = connected components
-connected components가 각각이 모두 open subset이고 top n-mnf가 된다.
-connected components의 개수가 at most countable
-for any x in top n-mnf, |FHG(top n-mnf,x)|:countable
-Smooth Manifold관련(Topological Manifold+Smooth Structure)
-T4
-second-countable
-paracompact
-homeomorphic인데 not diffeomorphic인 mnf들이 존재한다. 즉 mnf를 구분 짓는데에 smooth structure도 필요
-smooth chart는 모두 diffeo이다.(공역을 치역으로 제한하면)
-function on mnf(M:mnf)
-C^inf(M)은 VS(R(std))이다.
-for smooth f:M1->M2, pb(f):unital homor가 된다.
-(Fundamental Theorem for Line Integral of cvf)for f in C^inf(M), r:[a,b]->M, piecewise smooth curve in M일 때
lint over r df = f(r(b)) - f(r(a))
-(Existence of Smooth Partitions of Unity)for any open cover {E_i}, te a partition of unity {f_i} on TS dominated by {E_i}
(적당한 C^inf(M) 원소 잡을 때 쓰임)
-(Existence of Bump Function)for any U:open in M, A:closed in U, te f:smooth bump function for A supported in U(Partitions of unity쓰면 됨)
-(Extension Lemma)for any U:open in M, A:closed in U, f:A->R^k(std), smooth일 때
te F:M->R^k(std) s.t. restriction of F on A = f and support of F < U
-(Inverse Function Theorem for Manifolds)
(가정을 smooth말고 C^1으로 줄일 수도 있을 듯)
:f:M1->M2가 smooth이고 pf_p(f)가 bijective이면 te nbd(p) and nbd(f(p)) s.t. f|nbd(p)는 diffeo 게다가 pf_(f(p))(f^(-1))=(pf_p(f) )^(-1)
(만약 g:M1->M2, smooth, immersion and submersion이면 g:local diffeo를 알 수 있다.)
(pf가 원래 smooth map에 영향을 주는 예가 된다.)
-(Rank Theorem for Manifolds)
:for dim(M1)=m1, dim(M2)=m2, f:M1->M2가 smooth with constant rank k
for each x in M1, te smooth charts (V1,g1) for M1 centered at x and (V2,g2) for M2 s.t. (g2 o F o g1^(-1))((x1,x2,...,xm))=(x1,x2,...,xk,0,0,...,0)
(즉 smooth map이 아주 간단해짐, 그리고 image가 rank-dimensional, 이 theorem때문에 rank가 의미가 있는 것)
(마찬가지로 pf가 원래 smooth map에 영향을 주는 예가 된다.)
-for smooth f:M1->M2, M1:connected, TFAE
-for any p in M1, te smooth charts (V1,g1) containing p, (V2,g2) containing f(p), s.t. (g2 o F o g1^(-1))((x1,x2,...,xm))=(x1,x2,...,xk,0,0,...,0) for some k
-F has constant rank
-for smooth f:M1->M2, S:embedded submanifold of M2, Im(f)<S이면 f:M1->S도 smooth
-for smooth f:M1->M2 with constant rank
-f가 surjective이면 f는 submersion
-f가 injective이면 f는 immersion
-f가 bijective이면 f는 diffeomorphism
-(Equivariant Rank Theorem)
:M1:transitive smooth LG-space, M2:smooth LG-space, F:M1->M2 smooth, equivariant이면 F has constant rank,
그리고 level set은 closed embedded submanifolds of M1
-submersion관련(f:M1->M2가 submersion)
-product of mnfs->mnf, projection과 srv-bundle의 map p가 대표 submersion예
-closed under composition
-f:open map
-every p in M1 is in the image of a smooth local section of f
-if f:surjective, then f:quotient map
-for g:M2->M3, if f:surjective submersion, then g:smooth iff g o f:smooth
-for f:surjective submersion, if g:M1->M3:smooth, constant on the fibers of f, then te! h:M2->M3 s.t. h:smooth and h o f = g
-(uniqueness of smooth manifold quotient by surjective submersion)
:for f1:M1->M2, f2:M1->M3, 둘다 surjective submersions s.t. constant on each other's fibers, te! g:M2->M3 s.t. g:diffeo, g o f1 = f2
-immersion관련
-smooth curve f:interval->mnf with f'(t):nonzero for all t in interval이 대표 immersion예
-closed under composition
-f:M1->M2가 immersion이면 locally embedding이다.(for any p in M1, te nbd(p) in M1 s.t. restriction of f onto nbd(p):smooth embedding)
-smooth embedding관련
-inclusion:mnf->product of mnfs가 대표 smooth embedding예
-for smooth embedding f:M1->M2, M1 diffeo f(M1)
-f:M1->M2가 injective immersion이라해서 smooth embedding되지 않는다.(link)
(M1이 compact or f:proper란 조건이 붙으면 smooth embedding됨)
-closed under composition
-function on C^inf(M)(M:mnf, p in M)
-for smooth f:M1->M2, pb(f):C^inf(M2)->C^inf(M1), pb(f)(g)=g(f))
-tgs_p(M):VS(R(std))
-tgs_p(M), tv_p의 성질
-for any constant f in C^inf(M), tv_p(f)=0(link)
-for any f, g in C^inf(M) s.t. f(p)=g(p)=0, tv_p(fg)=0(link)
-(tgs_p is purely local)for any tv_p in tgs_p(M), any f, g in C^inf(M) s.t. f=g on a nbd(p) 일 때 tv_p(f)=tv_p(g)(link)
-(tgs_p(open submnf) iso tgs_p(M) as VS(F))U:open submanifold이고 i:inclusion of U일 때, pf_p(i)는 isomorphism (as VS(F)) for any p in U(link)
-f:M1->M2가 local diffeo면 pf_p(f)는 isomorphism for any p in M1
-따라서 f:submersion and immersion도 됨
-for V:f-dim VS(R(std)), any p in V, te F:isomorphism from V to tgs_p(V)
s.t. for any g in LT(V,W), pf_p(g) o F = G o g where G:isomorphism from W to tgs_g(p)(W)(즉 commute하게)
(따라서 V:f-dim VS(R(std))의 경우 일단 a in V를 택하면 for v in V, v can be identified with D_(*,v)(a) as derivation)
-(Chacracterization of tgs_p(M), using curves)
every tgv_p is the tangent vector to some smooth curve in M
(따라서 tgv는 derivation으로 이해하나, 혹은 M상의 smooth curve의 접vector로 이해하나 가능)
-(tgs of product mnfs)tgs_(p,q)(M1xM2) iso tgs_p(M1) x tgs_q(M2) as vector space
-function on tgs_p(M)(M:mnf)
-pf_p(f)의 성질(f:M1->M2, smooth, p in M1, g:M2->M3, smooth)
-pf_p(f):tgs_p(M1)->tgs_f(p)(M2)
-for g:tv_p, h in C^inf(M2), pf_p(f)(g)(h)=g(h(f)))
-pf_p(f):linear
-pf_p(g o f)=pf_f(p)(g) o pf_p(f)
-if id:identity on M, then pf_p(id)=identity on tgs_p(M)
-if f:diffeo, then pf_p(f):isomorphism from tgs_p(M1) to tgs_f(p)(M2)
-for any smooth chart (U,g), pf_p(g):tgs_p(M)->tgs_g(p)(R^n(std)), diffeo이다.
(따라서 for p in M, tgs_p(M)의 basis는 tgs_g(p)(R^n(std))의 basis의 inverse로 사용)
(for p in M, ctgs_p(M)의 basis는 위의 basis의 dual basis로 사용)
-pf(f)는 smooth bundle이 된다. TM->TN
-실질적인 계산 관련
-tgs_p(M)의 basis는 p에서의 coordinate system(즉 smooth chart)를 잡으면 해결됨(link)
-chart를 2개를 잡았다면(link)
-f:M1->M2 smooth, pf_p(f)는 p에서의 coordinate system과 f(p)에서의 coordinate system을 잡으면 pf_p(basis의 원소)가 어떻게 적히는 지 앎(link1)(link2)
-ctgs_p(M)의 basis는 p에서의 coordinate system를 잡으면 해결됨, chart를 2개 잡았을 때도 참고(link)
-for f in C^inf(U), df_p의 coordinate(link)
-submanifold관련(M:n-mnf, E:a subset of M)
-embedded k-submanifold관련
-if for some k, every p in E has nbd(p) in M s.t. nbd(p)교E:embedded k-submanifold of nbd(p), then E:embedded k-submanifold of M
(UOn이 embedded n-submanifold of R^(n+1)(std)임을 보일 때 사용됨, 즉 local에서 embedded submanifold만족하면...전체 subset도 된다는 것)
-E:embedded k-submanifold일 때, E:k-mnf(top k-mnf with subspace top, inclusion map E->M가 smooth embedding되게 smooth structure가짐)
(역 성립, smooth embedding의 image는 embedded submanifold가 된다.)
(위의 2 내용을 요약하면 embedded submanifolds are precisely the images of smooth embeddings)
-inclusion:E->M생각하면, pf_p(inclusion):tgs_p(E)->tgs_p(M), injective linear이니까 tgs_p(E) can be viewed as subspace of tgs_p(M)
-(Characterization tgs_p(E) as a subspace of tgs_p(M))
:for E:embedded submanifold and x in E, tgs_p(E)={X in tgs_p(M) s.t. Xf=0 for any f in C^inf(M) and f=0 on E}
-(Construct embedded submanifold, Graph)
:if U:open in R^n, F:U->R^k가 smooth이면 then the graph of F는 embedded n-dimensional submanifold of R^(n+k)
-(Constant-Rank Level Set Theorem)
:f:M1->M2 smooth with constant rank k일 때 level set of f는 closed embedded submanifold of codimension k.
-(Regular Level Set Theorem)
:f:M1->M2 smooth일 때 every regular level set은 closed embedded submanifold whose codimension is equal to the dimension of the range.
-(Characterization of embedded submanifold)
:E:embedded k-submanifold iff every p in E has a nbd(p) in M s.t. E교nbd(p) is a level set of a submersion F:nbd(p)->R^(n-k)(std)
-embedded submanifold 판정법
-정의대로
-image of smooth embedding으로써
-graph으로써
-level set으로써
-immersed k-submanifold관련
-immersed submanifolds are precisely the images of injective immersions.
-(Characterization tgs_p(E) as a subspace of tgs_p(M))
:embedded submanifold처럼 가능, 왜냐하면 smooth immersion의 image이므로
-covering관련(M:mnf(n-mnf), f:M1->M2 smooth covering map)
-f:immersion and submersion
-f가 injective이면 diffeo이다.
-(local continuous section of f의 존재성)for any x in M1, te nbd(f(x)) and g s.t. g:nbd(f(x))->M1, conti, f o g =id_nbd(f(x))
-for any M3, g:M2->M3:smooth iff g o f:smooth
-g:M1->M2가 proper local diffeo이면 g는 smooth covering map이다.
-if g:M1->M2가 topological covering map then M1:top n-mnf and M1 has a unique smooth structure s.t. g:smooth covering map.
-Complex Manifold관련
-any connected open subset of a Riemann surface is a Riemann surface
-Riemann surface is 2-dim C^inf manifold로 간주 될 수 있다.(여기서 2-dim 은 over R)
(왜냐하면 모든 holomorphic은 analytic이고 따라서 f(z)=u(z)+iv(z)를 f(x,y)=(u(x,y),v(x,y))로 보면 f:holo이면 f는 C^(inf))
-Every Riemann surface is orientable(transition map들이 holomorphic이고 derivative가 nonzero이므로 conformal되니까)
-Every Riemann Surface is an path-connected 2-dimensional C^inf real manifold
-Every compact Riemann surface is diffe to the g-holed torus, for some unique integer g>=0(g를 topological genus라 한다.)
-Lie Group관련
-EDP(LGi)도 LG(연산은 componentwise)
-LG:parallelizable(Lie(LG)의 basis생각)
-act_M by LG관련
-LG:discrete일 때 M:smooth LG-space iff for any g in LG, act_M by g:smooth
(즉 discrete LG action경우 smooth 판정이 쉽다.)
-LG-space관련
-for any g in LG, act_M by g는 homeo
-f:LG1->LG2, homoLG일 때 LG1:smooth LG1-space(left multiplication), 이 action과 f로 act_LG2 by LG1가능, 그러면 f는 equivariant됨(link)
-smooth LG-Space관련
-for any g in LG, act_M by g는 diffeo
-for LG, LG itself smooth LG-space(left multiplication)
-Lie(LG)관련
-Lie(LG):Lie subalgebra of VF(LG)
-Lie(LG) isomorphic tgs_e(LG) as vector spaces(link)
-dim(Lie(LG))=dim(LG)
-LG:abelian이면 Lie(LG):abelian
-f:LG1->LG2, homoLG이면 g:Lie(LG1)->Lie(LG2), lie algebra homo를 얻는다.
-vf,g(vf)는 f-related가 된다.
*Metric Space
-Space, Subspace, Subset관련
-(MetricS,d)에서의 성질
-any subset E is the intersection of open sets(countable intersection아닐 수 있음)
(주의:U_n := Union ball(x,1/n) for x in E, Intersection U_n is not E, but cl(E))
-any open set is an countable union of an increasing closed sets
-any closed sets is an countable intersection of an decreasing open sets
-First-Countability
-CGT
-T6
-(MetricS,d):totally bdd이면
-bdd이다.
-totally bdd under d iff totally bdd under d_sb
-모든 compact set은 bdd and closed
(역은 성립안함)
-compact=limit point compact=sequentially compact
-seq {x_n} in (MetricS,d)가 cauchy이면 te subseq s.t. d(x_n_k+1,x_n_k)<=2^(-k)(link)
-seq {x_n} in (MetricS,d)가 cauchy이고 subseq가 cv이면 seq {x_n}도 cv(link)
-CMetricS의 성질
-(Baire Category Theorem in CMetricS)X:CMetricS이면 X:Baire(link)
-C=a collection of f:CMetricS->R(std), conti이고 for any x in X, te M_x s.t. |f(x)|<=M_x for all f in C라면
te open U in X and te M s.t. |f(x)|<=M for all f in C and all x in U(link)
-complete is not top property
-CMetrics의 closed S도 CMetricS됨
-(MetricS,d)가 complete iff (MetricS,d_sb)가 complete
-(MetricS,d)가 complete iff every cauchy seq {x_n} in MetricS has a cv subseq.
-(MetricS,d)가 complete iff every nested seq {E_n} of nonempty closed subsets s.t. diam(E_n)->0, the intersection of E_n is nonempty.
-for any MetricS, te isom(MetricS,S of completion of MetricS), uniquely up to isom
-(Banach Fixed Point Theorem)
:CMetricS의 complete subset E, f:contraction on E일 때, f는 fixed point을 유일하게 갖고, iteration으로 얻어진다.
-KMetricS의 성질(KMetricS,d)
-(Heine-Borel Theorem)(MetricS,d):compact iff (MetricS,d):complete and totally bdd
-(Lebesgue Number Lemma)for any open covering, te delta>0 s.t. diam(E1)<delta이면 te E2 in the covering s.t. E1<E2(link)
-compact이므로 lindelof
-lindelof인데 metric이므로 second-countable
-complete
-LKT2, KT2의 성질들 모두 만족
-Metric관련
-metric d is conti(link)
-metric d가 induce한 top은 d가 conti가 되게하는 the smallest top이다.
-metric d와 d_sb는 같은 topology를 induce한다.
-d(x,E)=0 iff x is in cl(E)
-d(x,K)=d(x,a) for some a in K
-d(E1,E2)=0, E1:closed, E2:closed, E1,E2:disjoint일 수 있다. (R^2(std)에서 xy=1과 x축 생각)
-{x|d(x,E)<eps}=union of the open balls {x|d(a,x)<eps} for a in E(따라서 open)
-d(x,E):TS->R>=0 is conti
-K<open1이면, te open2 s.t.K<open2<open1 and open2={x|d(x,K)<eps}
-F1과 F2가 disjoint closed인데 d(E1,E2)=0일 수도 있다.
-isom(MetricS1,MetricS2)는 imbedding이고 따라서 isometric imbedding이라 하기도 함
-diam(E)의 성질
-monotone
-E1교E2 is not empty 이면 diam(E1UE2) <= diam(E1)+diam(E2)
-diam(E)=diam(cl(E))
-Continuous, Map관련
-MetricS에서의 conti criteria
-f:(MetricS1,d1)->(MetricS2,d2) conti <-> eps-delta definite이용((MetricS2,d2)가 R(std)일 때 주로 도움)
-f1:TS->R(std), f2:TS->R(std), f1 + f2, f1 - f2, f1 * f2, f1 / f2 모두 conti
-(Characterization of Closed Graph)(link)
:f:(MetricS1,d1)->(MetricS2,d2) has a closed graph iff if {x_n}:cv to x in MetricS1 and {f(x_n)}:cv to y in MetricS2 then y=f(x).
-compact metric space의 성질(KMetricS,d)
-(Uniform Continuity Theorem)f:KMetricS->MetricS가 conti이면 uni conti
-Functions collection관련(혹은 functions seq관련)(Range에 Metric이 있는 경우)
(Functions collection관련 in topology도 참고)
-for U:open in TS, A:closed in U, f:TS->R(std):conti, f:bump function for A supported in U란, 0<=f<=1 on TS, f=1 on A, support of f < U
-fC(J,MetricS)에서(uni cv, pt bdd, d_uni정의가능)
-fC(J,CMetricS)는 CMetricS됨 using d_uni(link)
-(d_sup)_sb=d_uni, when d_sup이 정의될 때
(d_sup이 정의되면, d_sup으로 논하는게 마음 편함, d_uni는 복잡함)
(즉, d_uni(f1,f2)=min{d_sup(f1,f2),1})
-fC(TS,MetricS)에서(top of K cv정의가능)
-top of pt cv < top of K cv < uni top
-seq {f_n} in fC(TS, MetricS) cv in the top of K cv iff for any K in TS, {f_n}:uni cv on K
-TS=K일 때
-top of K cv = uni top
-TS=discrete일 때
-top of pt cv = top of K cv
-fCbdd(J,MetricS)에서(d_sup정의가능)
-fCbdd(TS,MetricS)에서
-closed in fC(TS,MetricS) with uni top
-fCbdd(TS,CMetricS)는 CMetricS됨 using d_uni
-fCconti(TS,MetricS)에서(equiconti정의가능)
-KG-top = top of K cv
-(Uniform Limit Theorem)closed in fC(TS,MetricS) with uni top
-fCconti(TS,CMetricS)는 CMetricS됨 using d_uni
-E가 totally bdd using d_uni이면 E는 equiconti
-(Ascoli's Theorem)E가 equiconti, for any a in TS, E_a:={f(a) s.t. for some f in E}:pre-K in MetricS이면
-te S of fCconti(TS,MetricS) with top of K cv s.t. E<S, S:compact
-TS=CGT일 때
-closed in fC(TS, MetricS) with top of K cv
-{f_n}:cv in the top of K cv to f이면 f도 conti
-TS=LKT2일 때
-E<S, S:compact in fCconti(TS,MetricS) with top of compact cv이면 E는 equiconti이고 for any a in TS, E_a:pre-K in MetricS
-TS=K일 때
-MetricS=KMetric일 때, E가 equiconti이면 E는 totally bdd
-MetricS:all closed, bdd subsets are compact일 때,
-E:pre-K iff E:pt bdd, equiconti
-MetricS=R(std)일 때
-(Dini's Theorem)seq {f_n} in fCconti(TS,R(std))가 monotone, pt cv, limit f is conti이면 f_n은 uni cv
(유사하게, seq {f_n} in fC(K in R(std), R(std)), 각각이 monotone(conti일 필요 없음), pt cv to f which is conti on K이면 f_n은 uni cv도 된다.)
-MetricS=R^n일 때
-E:compact iff E:closed, bdd, equiconti
-(Arzela's Theorem){f_n} in fCconti(K,MetricS), {f_n}:pt bdd, equiconti이면 {f_n}은 uni cv인 subseq을 갖는다.
-fCcontibdd(TS,MetricS)에서
-fCcontiV(TS,R(std))에서
-E of fCcontiV(TS,R(std)), E:pre-K iff E:pt bdd, equiconti, vanishes uniformly at infinity
-fCcontiV(TS,C)에서
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