# Proof Verification: Only contractible discrete space is the one point space

I am trying to prove that only contractible discrete space is the one-point space. The proof of it can be seen here show that discrete space of $$X$$ is contractible if have one point and here Proving that a three-point discrete space is not contractible.

I believe I have 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 a 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 contractible.

Thanks!

• You ''secretly" used the connectivity of $[0, 1]$ for the fact "path-connected $\implies$ connected" which was necessary to reach a contradiction.
– 0XLR
Jan 16 at 4:45
• @OXLR Thanks so much!
– Phil
Jan 16 at 4:47