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Index basis (topology), Ôç injective, ç basis (vector space), ç interior, Ôò isomorphic, ¥ ∞ C , Ôý, ÔÔ isomorphism, ç Ck, Ôý, ÔÔ change of variables (integration), ÔÔ kernel, ç closed, Ôò closed form, Þ linear, ç closure, Ôò linear combination, ç coboundary, Þ linearly independent, ç coboundary map, Þ nullspace, ç cochain complex, Þ cochain homotopic, open cover, Ôò cochain homotopy, open set, Ôý cochain map, Þ cocycle, Þ partial derivative, Ôý cohomology, Þ compact, Ôò quotient space, â component function, À range, ç continuous, Ôç rank-nullity theorem, coordinate function, Ôý relative topology, Ôò dierential, Þ resolution, ¥ dimension, ç resolution of the identity, À direct sum, second-countable, Ôç directional derivative, Ôý short exact sequence, discrete topology, Ôò smooth homotopy, dual basis, À span, ç dual space, À standard topology on Rn, Ôò exact form, Þ subspace, ç exact sequence, ¥ subspace topology, Ôò support, Ô¥ Hausdor, Ôç surjective, ç homotopic, symmetry of partial derivatives, ÔÔ homotopy, topological space, Ôò image, ç topology, Ôò induced topology, Ôò total derivative, ÔÔ Ô trivial topology, Ôò vector space, ç well-dened, â ò Glossary Linear Algebra Denition. A real vector space V is a set equipped with an addition operation , and a scalar (R) multiplication operation satisfying the usual identities. + Denition. A subset W ⋅ V is a subspace if W is a vector space under the same operations as V. Denition. A linear combination⊂ of a subset S V is a sum of the form a v, where each a , and only nitely many of the a are nonzero. v v R ⊂ v v∈S v∈S ⋅ ∈ { } Denition.Qe span of a subset S V is the subspace span S of linear combinations of S. Denition. A subset S V is linearly⊂ independent if ( ) ⊂ av v 0 each av 0. v∈S ⋅ = Ô⇒ = Denition. A subset S V is a basisQ if it is linearly independent and spans V. We assume the axiom of⊂ choice. Every vector space has a basis. Denition. If a vector space has a nite basis of size n, we say that dim V n. Otherwise, we say that V is innite-dimensional. = is makes sense because if dim V n, then all bases have size n. If W V is a subspace, and if dim W dim V, then V W. = Denition.⊂ A map L V W between= two vector spaces= V and W is linear if ∶ → L αv1 βv2 αL v1 βL v2 . Denition. e kernel or nullspace( of a+ linear) map= L( V) + W( is) the subspace ker L V dened by ker L v V L∶ v → 0 . ⊂ Denition. A linear map L V W is( injective) = { ∈if kerS (L ) 0.= } Denition. e image or range∶ →of a linear map L V =W is the subspace image L W dened by image L L v∶ W→ v V . ( ) ⊂ Denition. A linear map L V W is(surjective) = { ( if) image∈ S L∈ }W. Denition. A linear map L ∶ V→ W is an isomorphism if( there) = exists a linear map L−1 W V −1 −1 such that L L IdV and LL IdW . ∶ → ∶ → = = ç A linear map is an isomorphism i it is both injective and surjective. Denition. Vector spaces V and W are isomorphic if there exists an isomorphism from V to W. Note that V and W are isomorphic i dim V dim W. Denition. A sequence of linear maps = L−2 L−1 L0 L1 L2 L3 V−1 V0 V1 V2 V3 is exact if ker Li imageLi−1 ⋯Vi→. A nite→ sequence→ → of the→ form→ ⋯ = ⊂ L0 L1 L2 Ln−1 V0 V1 V2 Vn is exact if this condition holds wherever→ it makes→ sense,→ ⋯ i.e.→ for i 1, . , n 1 . Denition. Given a subspace W V0, a resolution of the subspace∈ { W is an− exact} sequence of the form ⊂ L0 L1 L2 Ln−1 Ln V0 V1 V2 Vn 0 such that ker L W. 0 → → → ⋯ → → Example. By the= Poincaré lemma, the de Rham complex of a star-shaped region U Rn is a reso- lution of the subspace of constant functions R Ω0 U . ⊂ eorem. If ⊂ ( ) L 0 V1 V2 is exact, then L is injective. → → Proof. ker L V0 0. = = eorem. If L V1 V2 0 is exact, then L is surjective. → → Proof. imageL ker V2 0 V2, so L is surjective. = ( → ) = Corollary. If L 0 V1 V2 0 is exact, then L is an isomorphism. → → → Proof. Since L is both injective and surjective, L is an isomorphism. ¥ Denition. A short exact sequence is an exact sequence of the form L1 L2 0 V1 V2 V3 0. eorem. For any short exact sequence of→ nite-dimensional→ → → vector spaces, dim V2 dim V1 dim V3. Proof. is proof may seem elaborate, but it follows the usual recipes of linear= algebra.+ Suppose dim V n. en we may take a basis e n of V . us L e n span the image of L . Since 1 i i=1 1 1 i i=1 1 n r n L1 is injective, L1 e are independent in V2. We can nd vectors f so that L1 e i i=1 j j=1 i i=1 r = { } { ( )} n f becomes a basis of V2. us dim V2 n r. Note that L1 e image L1 ker L2 , j j=1 { ( )} i i=1 { ( )} ∪ r so L2 L1 e 0. Now L2 is surjective, so the L2 f must span V3. If we can show that i = + j j=1 { ( )} ⊂ ( ) = ( ) r L2 f are independent, then dim V3 r, and the theorem follows. (j (j=1 )) = ( ) To show( ) independence, suppose a jL2 f=j 0. en 0 L2 a j f j a j f j ker L2 image L . Since L e n span the image of L , we have 1 1 i i=1 1 ∑ ( ) = = ∑ Ô⇒ ∑ ∈ = ( ) { ( )} r n a j f j bi L1 ei , j=1 i=1 Q = Q ( ) for some constants b n . In particular, i R i=1 { ∈ } n r bi L1 ei a j f j 0. i=1 j=1 Q − ( ) + Q = n r Since L1 e f are independent in V2, all the coecients b and a must be zero. i i=1 j j=1 i j We wanted to show that all the a j 0, so we are done. { ( )} ∪ { } eorem. If L V W, then = ∶ → 0 ker L i V L image L 0 is a short exact sequence, where i ker→ L V→ is the→ inclusion( ) map.→ Corollary (Rank-nullity). If L V∶ W→ is a linear map of nite-dimensional vector spaces, then ∶ dim→ V dim ker L dim imageL. Denition. e direct sum of V W of two= vector spaces+ is the vector space of pairs v, w v V, w W , where the operations are v1, w1 v2, w2 v1 v2, w1 w2 and α v1, w1 αv1, αw1 . ⊕ {( )S ∈ ∈ } eorem. If A, B V are( subspaces) + ( such that) ∶= V( span+ A +B ,) then ( ) = ( ) ⊂ 0 A B =A B ( V∪ )0 is a short exact sequence, where A B → A ∩B is→ the⊕ diagonal→ map→ v v, v , and the map A B V is the dierence map a, b a b. ∩ → ⊕ ↦ ( ) ⊕ → ( ) ↦ − Corollary. dim V dim A dim B dim A B . Denition. If W V is a vector subspace, then the quotient space is the set of equivalence classes = + − ( ∩ ) V W v v V and v v v v W . ⊂ 1 2 1 2 e equivalence classes~ are∶= thus{[ of] S the∈ form v[ ]v= [W], i.e.⇐⇒ translates− of∈ the} subspace W. Example. Suppose W x , 0, 0 3 x . en 3 W x , x , x where we ignore 1 R 1[ ]R= + R 1 2 3 the x1-coordinate. e x1-coordinate is still there, but it gets ignored. For example, ∶= {( ) ∈ S ∈ } ~ = {[( )]} 5, 3, 7 0, 3, 7 . We could always set the x -coordinate to zero, but we don’t have to. 1 [( )] = [( )] eorem. If W V is a subspace, then i q ⊂ 0 W V V W 0 is exact, where i is the inclusion map, and q is the quotient map q v v . → → → ~ → Corollary. ( ) = [ ] dim V W dim V dim W. Suppose that X and Y are sets, and that X~ has= an equivalence− relation . Given an element x X, let x denote its equivalence class. Let X denote the “quotient set” of equivalence classes. ∼ ∈ Denition. A function f X Y is well-dened on the quotient set if [ ] ~ ∼ f x f x whenever x x . ∶ → 1 2 1 2 When f is well-dened on X , it( makes) = ( sense) to dene a function∼ f X Y according to the rule ~ ∼ f x f x . ∶ ( ~ ∼) → Example. Suppose V1 and V2 are vector spaces, and W V1 is a subspace. In order to dene a linear map ([ ]) ∶= [ ( )] V1 W V2⊂, it suces to dene a linear map ~ → L V1 V2 and show that L is well-dened on the quotient space. We must check the condition ∶ → L v1 L v2 whenever v1 v2 L v1 v2 0 whenever v1 v2 W ( ) = ( ) ∼ L w 0 whenever w W ⇐⇒ ( − ) = − ∈ W ker L. ⇐⇒ ( ) = ∈ Since W is the subspace of elements which represent zero in the quotient, ⇐⇒ ⊂ a linear map is well-dened i everything which represents zero actually maps to zero. Whenever this holds, it makes sense to dene L V1 W V2 by L v L v . ∶ ( ~ ) → ([ ]) ∶=â [ ( )] Chain complexes Denition. A sequence of linear maps p−2 p−1 p p+1 p+2 p+3 d C p−1 d C p d C p+1 d C p+2 d C p+3 d is called a cochain complex⋯ →if image d→p−1 →ker d p →C p.
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  • General Topology

    General Topology

    General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry).