Proving Homotopy Equivalence and Contractibility in Algebraic Topology

[Flying_Goat]

Hi, I am stuck on two problems from Allen Hatcher's book, Algebraic Topology.

Homework Statement
4. A deformation retraction in the weak sense of a space X to a subspace A is a homotopy $f_t$: X→X such that $f_0$=$1_X$ (the identity map on X), $f_1$(X) $\subset$ A, and $f_t (A) \subset A$ for all t. Show that if X deformation retracts to A in this weak sense, then the inclusion $i_A:$A→X is a homotopy equivalence.

The attempt at a solution
We want to show that there exists f:X→A such that $f i_A$ $\approx$ $1_A$ and $i_A f \approx 1_X$. Define F:A$\times$I→A by F(t,x) = $f_t i_A$. We then have F(0,x) = $f_0 i_A$ = $1_A$ and F(1,x)=$f_1 i_A$.

Now, for each t $\in$ [0,1] define $X_t=f_t(X)$. Let $i_{X_t} : X_t → X$ be the natural inclusion. Now, define G:X$\times$I→X to be G(t,x) = $i_{X_t} f_t$. Then G(0,x) =$i_X f_0 = 1_X$ and G(1,x) = $i_{X_1} f_1$. Since $f_1(X)\subset A$, it follows that $i_{X_1} f_1 = i_A f_1$.

I don't know how to prove that F is continous. I am not sure if G is even continous, its the only thing i could come up with. Also I haven't used the fact that $f_t (A) \subset A$ for all t. What did I miss?

Homework Statement
9. Show that a retract of a contractible space is contractible.

The attempt at a solution
So we are given a retract from a space X to a subset A. We know that there exists a homotopy between $1_X$ and a constant map. I am not sure where to go from here. I just wanted to ask that if say X contracts to $x_0$, does A necessarily contain $x_0$? Because if $x_0\in A$ then we can just restrict the homotopy to A and we would have shown that A is contractible.

Any help would be appreciated.

 

[lippyka]

For the first problem, you have the right idea, but you need to show that F and G are continuous. To do this, you can use the fact that f_t (A) \subset A for all t. Show that for any x in A, F(t,x) is continuous in t and G(t,x) is continuous in t.For the second problem, the key idea is to show that the retract r:X→A is actually a homotopy equivalence. This means that there exists a homotopy h:A→X such that hr≅1_A and r h≅1_X. Since X is contractible, this implies that A is contractible as well.