ABSTRACT
Dimensions 143
3.5.7 Linear Shape Spaces and Affine Shape Spaces as
Grassmannians 143
3.5.8 Projective Shape Spaces 146
3.6 Exercises 149
3.1 Manifolds, Submanifolds, Embeddings, Lie Group actions
A topological space is a pair (M,τ), whereM is a nonempty set, and τ is a set of parts of M, such that (i) M is in τ, (ii) for any finite set of parts in τ, their intersection is still in τ, and (iii) for any set of parts in τ, their union is in τ. A set in τ is said to be open, and its complement is said to be a closed set . A function f : (M1,τ1)→ (M2,τ2) between two topological spaces is continuous if ∀W ∈ τ2, f−1(W ) ∈ τ1. If in addition f is one to one and its inverse is continuous, f is said to be a .
