R^{n} a vector space over R (or C) with canonical basis {e_{1}, ..., e_{n}} where e_{i} = (0, ., 0, 1, 0, ..., 0) Inner product on R^{n}: (x, y) = Σ_{i=1}^{n}x_{i}y_{i} The basis is orthonormal: (e_{i}, e_{j},) = δ_{ij} = \{ \begin{matrix}0 & i ≠ j \\ 1 & i = j \end{matrix}

d(x, y) = ||x − y|| = (x, y)^{\frac{1}{2}} The norm of x = ||x|| = d(x, 0) B_{ϵ}^{n}(x) = {y ∈ R^{n} | d(x, y) < ϵ} = ball of radius ϵ centered at x. C_{ϵ}^{n}(x) = {y ∈ R^{n} | |x_{i} − y_{i}| < ϵ, i = 1, ..., n} = cube of side 2ϵ centered at x. R^{1} = R, R^{0} = {0} . I.2 R^{n} = E^{n} where a coordinate system is defined on E^{n} A property is Euclidean if is does not depend on the choice of an orthonormal coordinate system. I.3 Topological Manifolds Defn: M is locally Euclidean of dimension n if for all p ∈ M, there exists an open set U_{p} such that p ∈ U_{p} and there exists a homeomorphism f_{p} : U_{p} → V_{p} where V_{p} ⊂ R^{n}.

Defn 3.1: An n-manifold, M, is a topological space with the following 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 an n−manifold.

Ex 3.3: S^{n} = {x ∈ R^{n+1} | ||x|| = 1} is an manifold Proof. stereographic projection: projection:

Remark 3.5. For a “smooth” manifold, M ⊂ R^{n}, can choose a projection by using the fact that for all p ∈ M there exists a unit normal vector N_{p} and tangent plane T_{p}(M) which varies continuously with p.

Example: smooth and non-smooth curve.

Example 3.4: The product of two manifolds is also a manifold.

Example: Torus = S^{1} × S^{1}.

Theorem 3.6: A manifold is

1.) locally connected, 2.) locally compact, 3.) a union of a countable collection of compact subsets, 4.) normal, and 5.) metrizable.

Defn: X is locally connected at x if for every neighborhood U of x, there exists connected open set V such that x ∈ V ⊂ U. X is locally connected if x is locally connected at each of its points.

Defn: X is locally compact at x is there exists a compact set C ⊂ X and a set V open in X such that x ∈ V ⊂ C. X is locally compact 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 all closed sets B and for all points x ∉ 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 all pairs 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 R^{n} = R^{m}, then n = m.

Recall: M is locally Euclidean of dimension n if for all p ∈ M, there exists an open set U_{p} such that p ∈ U_{p} and there exists a homeomorphism f : U_{p} → V_{p} where V_{p} ⊂ R^{n}.

(U_{p}, f) is a coordinate nbhd of p.

Given (U_{p}, f) Let q ∈ U ⊂ M. f(q) = (f_{1}(q), f_{2}(q), ...., f_{n}(q)) ∈ R^{n} are the coordinates of q.

I.4 Manifolds with boundary and Cutting and Pasting If dimM = 0, then M = If M is connected and dimM = 1, then Thm 4.1: Every compact, connected, orientable 2-manifold is homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes Let upper half-space, H^{n} = {(x_{1}, x_{2}, ..., x_{n}) ∈ R^{n} | x_{n} ≥ 0}, ∂H^{n} = {(x_{1}, x_{2}, ..., x_{n−1}, 0) ∈ R^{n}} ∼ R^{n−1} M is a manifold with boundary if it is Hausdorff, has a countable basis, and if for all p ∈ U, there exists an open set U_{p} such that p ∈ U_{p} and there exists a homeomorphism f : U_{p} → V_{p} one of the following holds: i.) V_{p} ⊂ H^{n} − ∂H^{n} (p is an interior point) or ii.) V_{p} ⊂ H^{n} and f(p) ∈ ∂H^{n} (p is a boundary point). ∂M = set of all boundary points of M is an (n-1)-dimensional manifold.