Understanding the solution of a problem in section 2.1 in AT.

Question

The problem and its solution are given below:

My questions are:

1- why do we have to define $A$ by removing a point $a$ from the unreduced suspension $SX$ of $X$? is this because of that the definition of the upper cone excludes the point at the tip(it seems like I do not fully understand the topological definition of a cone because I think the geometric definition (if I understand correctly)of the cone does not exclude the point at the tip)?

2- can anyone explains to me by words (intuitively without any mathematical symbols) what makes the cone contractible?

Answer 1

1. To apply Mayer-Vietoris you need two subspaces such that their interiors cover whole space. So for example open subsets that cover the space. Now if we can pick $A$ and $B$ in such a way that they are additionally contractible then this is even better. Because then we have lots of zeros inside the Mayer-Vietoris sequence. Finally it is a nice property of $SX$ that $SX\backslash\{p\}$ is open and contractible if $p$ is one of the tips.
2. Cone has one special vertex: the top. You can contract the cone by moving every point to the top by a straight line.

This is cone:

Although in general topology the base can be any topological space, not only a disk. You contract it by moving (and shrinking) the base towards the top step by step.

And suspension is just two cones glued by their bases.

Here's an example of suspension of a disjoint union of two disc-like shapes (forgive me my horrible drawing):