Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 2 down vote accepted

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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.