By Dundas B.I.

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12 Given 1. a set M, 2. a collection A of subsets of M and 3. an injection xU : U → Rn for each U ∈ A, such that 1. there is a countable subcollection of A which covers M 2. for p, q ∈ M, either there is a U ∈ A such that p, q ∈ U or there are U, V ∈ A such that U and V are disjoint with p ∈ U and q ∈ V , and 3. if U, V ∈ A then the bijection xU (U ∩ V ) → xV (U ∩ V ) sending q to xV xU −1 (q) is a smooth map between open subsets of Rn . Then there is a unique topology on M such that (M, D({(xU , U)}U ∈A )) is a smooth manifold.

Their importance lies in the fact that they focus the attention on a neighborhood of p, ignoring everything “far away”. The existence of smooth bump functions is a true luxury about smooth manifolds, which makes the smooth case much more flexible than the analytic case. We will return to this topic when we define partitions of unity. 14 Let X be a space and p a point in X. A bump function around p is a map φ : X → R, which takes values in the closed interval [0, 1] only, which takes the constant value 1 in (the closure of) a neighborhood of p, and takes the constant value 0 outside some bigger neighborhood.

That a matrix has rank r means that it has an r × r invertible submatrix, but no larger invertible submatrices. r The subset Mm×n (R) ⊆ Mm×n R of matrices of rank r is a submanifold of codimension (n − r)(m − r). Since some of the ideas will be valuable later on, we spell out a proof. For the sake of simplicity, we treat the case where our matrices have an invertible r × r submatrices in the upper left-hand corner. The other cases are covered in a similar manner, taking care of indices (or by composing the chart we give below with a diffeomorphism on Mm×n R given by multiplying with permutation matrices so that the invertible submatrix is moved to the upper left-hand corner).

### Differential Topology by Dundas B.I.

by Richard

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