Rna vector space over R (or C) with canonical basis {e1, ..., en}where ei= (0, ., 0, 1, 0, ..., 0)Inner product on Rn: (x, y) = Σni=1xiyiThe basis is orthonormal: (ei, ej, ) = δij=0 i 6= j1 i = jd(x, y) = ||x − y|| = (x, y)12The norm of x = ||x|| = d(x, 0)Bnǫ(x) = {y ∈ Rn| d(x, y) < ǫ} = ball of radius ǫ centered at x.Cnǫ(x) = {y ∈ Rn| |xi− yi| < ǫ, i = 1, ..., n} = cube of side 2ǫcentered at x.R1= R, R0= {0} .I.2Rn= Enwhere a coordinate system is defined on EnA property is Euclidean if is does not depend on the choice of anorthonormal coordinate system.I.3 Topological ManifoldsDefn: M is locally Euclidean of dimension n i f for all p ∈ M, thereexists an open set Upsuch that p ∈ Upand th ere exists a homeo-morphism fp: Up→ Vpwhere Vp⊂ Rn.1Defn 3.1: An n-manifold, M, is a topological space with the follow-ing properties:1.) M is locally Euclidean of dimension n.2.) M is Hausdorff.3.) M has a countable basis.Give an example of a locally Euclidean space which is not Hausdorff:Ex 3.2: If U is an open subset of an n−manifold, then U is also ann−manifold.Ex 3.3: Sn= {x ∈ Rn+1| ||x|| = 1} is anmanifoldProof. stereographic projection:2projection:Remark 3.5. For a “smooth” manifold, M ⊂ Rn, can cho ose a pro-jection by using the fact that for all p ∈ M there exists a unit normalvector Npand tangent plane Tp(M) which varies continuously withp.Example: smooth and non-smooth curve.Example 3.4: The product of two manifolds is also a manifold.Example: Torus = S1× S1.Theorem 3.6: A manifold is1.) locally connected, 2.) locally compact, 3.) a union of a countablecollection of compact subsets, 4.) normal, and 5.) metrizable.Defn: X is locally connected at x if for every neighborhood U ofx, there exists connected open set V such that x ∈ V ⊂ U. X isloc ally connected if x is l ocally connected at each of its points.3Defn: X is locally compact at x is there exists a compact setC ⊂ X and a set V open in X such that x ∈ V ⊂ C. X is locallycompact if it is locally compact at each of its points.Defn: X is regular if one-point sets are closed in X and if for allclosed sets B and for all points x 6∈ B, there exist disjoint open sets,U, V, such that x ∈ U and B ⊂ V .Defn: X is normal if one-point sets are closed in X and if for allpairs of disjoint closed sets A, B, there exist disjoint open sets, U,V, such that A ⊂ U and B ⊂ V .Brouwer’s Theorem on Invariance of Domain (1911). If Rn= Rm,then n = m.Recall: M is locally Euclidean of dimension n if f or all p ∈ M ,there exists an open set Upsuch that p ∈ Upand there exists ahomeomorphism f : Up→ Vpwhere Vp⊂ Rn.(Up, f) is a coordinate nbhd of p.Given (Up, f) Let q ∈ U ⊂ M. f (q) = (f1(q), f2(q), ...., fn(q)) ∈ Rnare the coordinates of q.4I.4 Manifolds with boundary and Cutting and PastingIf dimM = 0, then M =If M is connected and dimM = 1, thenThm 4.1: Every compact, connected, orientable 2-manifold is home-omorphic to a sphere, or to a connected sum of tori, or to a connectedsum of projecti ve planes5Let upper half-space, Hn= {(x1, x2, ..., xn) ∈ Rn| xn≥ 0},∂Hn= {(x1, x2, ..., xn−1, 0) ∈ Rn} ∼ Rn−1M is a manifold with bound ary if it is Hausdorff, has a countablebasis, and if for all p ∈ U, there exists an open set Upsuch thatp ∈ Upand there exists a homeomorphism f : Up→ Vpone of thefollowing holds:i.) Vp⊂ Hn− ∂Hn(p is an interior point) orii.) Vp⊂ Hnand f(p) ∈ ∂Hn(p is a boundary point).∂M = set of all boundary points of M is an (n-1)-dimensional