I am trying to prove that only contractible discrete space is the one point space. The proof of it can be seen here(https://math.stackexchange.com/questions/2218782/show-that-discrete-space-of-x-is-contractible-if-have-one-point) and here(https://math.stackexchange.com/questions/1720201/proving-that-a-three-point-discrete-space-is-not-contractible)

I believe I have a proof, but I do not use connectedness of the unit interval, so I’m asking for verification of my proof.

My proof

Suppose discrete space X was contractible. Any contractible space is path-connected. Indeed, take a homotopy $H colon X times (0,1) to X$ from identity on $X$ to $cost$ on $x_1$. Then for any $x_2 in X$, $H(x_2, -) colon (0,1) to X$ gives a path from $x_1$ to $x_2$ as $H$ being continuous implies $H$ is continuous in each variable.

Now I claim that any discrete space with more than 2 points is not connected. To show this, note that if there is continuous and non-constant map $X to {0,1}$, then $X$ is not connected. As $X$ has more than 2 points, we can find a non-constant set map $X to {0,1}$. And as $X$ is discrete, this non-constant set map must be continuous.

So if $X$ has more than 1 point and contractible, we get that $X$ is path connected but not connected. This is impossible, so $X$ must have one point if it were to be contractilbe.

Thanks!