A Universe of Sorts

Two spaces have the same homotopy type iff there are functions $f: X \to Y$ and $g: Y \to X$ such that $f \circ g$ and $g \circ f$ are homotopic to the identity.
Now consider two spaces: (1) the point, (2) the topologists's sine curve with two ends attached (the warsaw circle).
See that the second space can have no non-trivial fundamental group, as it's impossible to loop around the sine curve.
So the warsaw circle has all trivial $\pi_j$ , just like the point.
See that the map $W \to \{ \star \}$ must send every point in the warsaw circle to the point $\star$ .
See that the map backward can send $\star$ somewhere, so we are picking a point on $W$ .
The composite smooshes all of $W$ to a single point. For this to be homotopic to the identity is to say that the space is contractible.

§ Spaces That Have Same Homotopy Groups but Not the Same Homotopy Type