We know that if a space $X$ has the property that $id_X$, the identity map on $X$, is null-homotopic, then $X$ is contractible.
My question is what happen to a space $X$ where the identity map on $X$ is NOT null-homotopic, why can't it be contractible? What goes wrong? Any examples?
Here contractible refers to a property of a space having the homotopy type of a point.
Could anyone help me clarify this confusion? Thanks.