I have a question about whether contractible spaces are path connected or not. I've been solving questions in my textbook, and I've used this many times to the point where I'm not even sure about it anymore.
Let $X$ be contractible. Then $X$ is homotopy equivalent to a one point space $\{*\}$. So there exists functions $f: X \to \{*\}$ and $g:\{*\} \to X$ such that $g\circ f \simeq id_X$.
Let $H: X\times I \to X$ be a homotopy such that $H(X, 0) =g\circ f $ and $H(X, 1) = id_X$. The function $g\circ f$ however is constant, so let $p = (g\circ f)(x)$, $\forall x \in X$. Suppose I had a point $a\in X$.
Is $H(a, t): I\to X$ a path in $X$ from $a$ to $p$? Is it always continuous? I used it to prove that if a space $X$ is contractible, then every point in $X$ is a deformation retract.
Let X be homotopy equivalent to the one point space $\{*\}$. Then $\exists$ $f: X \to \{*\}$ and $g:\{*\} \to X$ such that $f\circ g \simeq id_{\{*\}}$ and $g\circ f \simeq id_X$. In particular, let $H$ be the homotopy such that $H(X, 0) = id_X$ and $H(X, 1) = g\circ f = p\in X$.
Let $a \in X$. Define $r: X \to \{*\}$ by $r(x) = a$, $\forall x \in X$. Let $j: \{*\}\to X$ be the inclusion map. We need to show $j\circ r \simeq id_X$.
$H(a, t)$ is a path in X from $a$ to $p$. Define $F: X\times I \to X$ by $$F(x, t) = \left. \begin{cases} H(x, 2t), & \text{for } t\leq \frac{1}{2} \\ H(a, 2-2t), & \text{for }t \geq \frac{1}{2} \end{cases} \right\} $$
Then $F(X, 0) = id_X$ and $F(X, 1) = a=j\circ r$ is a homotopy between the $id_X$ and the retraction $j\circ r$.
I would appreciate any critiques.
Edit
After insights provided by William in the comments, this proof does not work if we require "Strong Deformation Retraction" or that a set $A\subset X$ be fixed throughout the homotopy for it to be a DR. I don't know if this can be proved to be false, I will leave it open until myself or someone comes up with a proof on why it is false if we require Strong DR.