Is there any deformation retraction between a point and a circle?

Question

A point is a retraction of every topological space but how it isn't a Deformation Retraction of s1(circle)?

Answer 1

A deformation retract is a homotopy equivalence, so in particular it induces isomorphisms on fundamental groups. $\pi_1(\mathrm{pt}) = 0$ but $\pi_1(S^1) = \Bbb Z$, so a point cannot be the deformation retract of a circle (or any other space with a nontrivial homotopy group).