Proposition

Show that the following are equivalent:

$X$ is contractible.
$X$ is homotopy equivalent to a one-point space.
Each point of $X$ is a deformation retract of $X$.

Solution

$1 \rightarrow 2$

Let $X$ be a contractible space. Then there exists a point $p \in X$ and a continuous map $H: X \times I \rightarrow X$ such that

$\forall x \in X, H(x, 0) = x$.
$\forall x \in X, H(x, 1) = p$.

Let $\phi: X \rightarrow \{ p \}$ map every element in $X$ to $p$. Let $\psi: \{ p \} \rightarrow X$ be the inclusion map.

$\phi \circ \psi$ is the identity map for $\{ p \}$. Thus it is indeed homotopic to the identity map for $\{ p \}$.
$\psi \circ \phi$ maps every point in $X$ to $p$. Then $H$ is a homotopy from the identity map for $X$ to $\psi \circ \phi$. Therefore, $\psi \circ \phi$ is homotopic to the identity map for $X$.

Thus $X$ is homotopy equivalent to $\{ p \}$, a one-point space.

$2 \rightarrow 3$

Suppose that $X$ is homotopy equivalent to a one-point space $Y = \{ y \}$. Let $x_0 \in X$. We will show that $\{ x_0 \}$ is a deformation retract of $X$.

Let $f_1: \{ x_0 \} \rightarrow \{ y \}, f_2: \{ y \} \rightarrow \{ x_0 \}$. Then $f_1 \circ f_2$ and $f_2 \circ f_1$ are both the identity maps on $\{ y \}$ and $\{ x_0 \}$, respectively. Thus $\{ x_0 \}$ is homotopy equivalent to $\{ y \}$.

Since homotopy equivalence is an equivalence relation, $X$ is homotopy equivalent to $\{ x_0 \}$.

Then there exists a continuous map $F: X \times I \rightarrow X$ such that

$F(x, 0) = x_0$ for all $x \in X$.
$F(x, 1) = x$ for all $x \in X$.

Let $r: X \rightarrow \{ x_0 \}$ be defined such that $r(x) = F(x, 0)$.

$r$ is a constant function, so it is continuous.
$r$ fixes $x_0$.

Thus, $r$ is a retraction. Moreover, $F$ is a homotopy from $i_{\{ x_0 \}} \circ r$ to the identity map of $X$. Therefore, $\{ x_0 \}$ is a deformation retract of $X$.

$3 \rightarrow 1$

Let $x \in X$ be given arbitrarily. Then $A = \{ x \}$ is a deformation retract of $X$. This implies the existence of a deformation retraction $r: X \rightarrow A$ such that $i_A \circ r$ is homotopic to the identity map of $X$. Since $A$ is a singleton, $r$ is the constant function that maps every point in $X$ to $x$. Thus the identity map of $X$ is homotopic to a constant map, so $X$ is contractible.