The first exercise given in Spanier's, Algebraic Topology is:
$X$ is contractible if and only if it is a retract of any cone over $X$.
I have proven the first implication, however I am stuck on the second implication.
In Spanier he defines a cone over a topological space $X$ with vertex $v$ as the mapping cylinder of the constant map $X\rightarrow v$. In addition he denotes the mapping cylinder as $Z_{f}$.
Starting with the second statement that $X$ is a retract of any cone over $X$, I have shown it is certainly a retract of $Z_{c}$ where c is the constant map from $X$ to $x_{0}$ for some $x_{0}\in X$ and I've also shown that $P$ can be imbedded as a weak deformation retract of $Z_{c}$, where $P$ is the one point space containing $x_{0}$.
Now in Spanier there is a theorem stating:
Two spaces $X$ and $Y$ have the same homotopy type if and only if they can be imbedded as weak deformation retracts of the same space $Z$.
Since I have shown $P$ can be imbedded as weak deformation retract of $Z_{c}$ and since $X$ is already a retract of $Z_{c}$ then it suffices to show $Z_{c}$ is deformable into $X$ to complete the proof.
So first I wanted to know if this is a good way of attacking the proof and if I haven't stated anything incorrectly? If so, then I wanted to know of any way to show $Z_{c}$ is deformable into $X$?
I just started this book and so my knowledge of algebraic topology is only up to the fourth section of Spanier.
Best Answer
I think you're trying too hard. Note that for any cone $CX$ over a point $v$, $CX$ deformation retracts onto $v$, and in particular is contractible. By hypothesis $X$ is a retract of a contractible space, so it must be contractible.