A space $X$ is contractible if and only if it is homotopically equivalent to the space with one point.

Question

I understand that for $\{pt\}\rightarrow X \rightarrow \{pt\}$ we can compose the inclusion map with the constant map and get the identity. However, for $X\rightarrow \{pt\} \rightarrow X$, I do not understand how we can get a continuous mapping from $\{pt\}$ to $X$ that, when composed with the constant map, give us the identity for all of $X$. Could someone convince me of this?

Answer 1

The definition of "homotopy equivalence" does not require the composition $X\to\{pt\}\to X$ to be the identity map; it only required it to be homotopic to the identity map. That is, there should be a homotopy from the identity map $X\to X$ to a constant map. This is exactly what a contraction gives you.