Algebraic Topology: Lecture 3

We first discussed the long-exact sequence of homotopy groups associated with a pair (X,A) of a space X and subspace A. We then introduced the notion of a CW-structure on a space. We gave the n-sphere two different CW-structures. The first CW-structure has one 0-cell and one n-cell; the second CW-structure has two i-cells for all i between 0 and i. The second CW-structure descends to a CW-structure with one i-cell for all i between 0 and n on the real projective n-space. Not every space can be given a CW-structure. But for every space X, there exists a weak equivalence X' -> X from a space X' that can be given a CW-structure. For reading, see Hatcher, Algebraic Topology, chapters 0 and 4.